/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker

! This file was ported from Lean 3 source module algebra.gcd_monoid.basic
! leanprover-community/mathlib commit baba818b9acea366489e8ba32d2cc0fcaf50a1f7
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Algebra.Associated
import Mathbin.Algebra.GroupPower.Lemmas
import Mathbin.Algebra.Ring.Regular

/-!
# Monoids with normalization functions, `gcd`, and `lcm`

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

This file defines extra structures on `cancel_comm_monoid_with_zero`s, including `is_domain`s.

## Main Definitions

* `normalization_monoid`
* `gcd_monoid`
* `normalized_gcd_monoid`
* `gcd_monoid_of_gcd`, `gcd_monoid_of_exists_gcd`, `normalized_gcd_monoid_of_gcd`,
  `normalized_gcd_monoid_of_exists_gcd`
* `gcd_monoid_of_lcm`, `gcd_monoid_of_exists_lcm`, `normalized_gcd_monoid_of_lcm`,
  `normalized_gcd_monoid_of_exists_lcm`

For the `normalized_gcd_monoid` instances on `ℕ` and `ℤ`, see `ring_theory.int.basic`.

## Implementation Notes

* `normalization_monoid` is defined by assigning to each element a `norm_unit` such that multiplying
by that unit normalizes the monoid, and `normalize` is an idempotent monoid homomorphism. This
definition as currently implemented does casework on `0`.

* `gcd_monoid` contains the definitions of `gcd` and `lcm` with the usual properties. They are
  both determined up to a unit.

* `normalized_gcd_monoid` extends `normalization_monoid`, so the `gcd` and `lcm` are always
  normalized. This makes `gcd`s of polynomials easier to work with, but excludes Euclidean domains,
  and monoids without zero.

* `gcd_monoid_of_gcd` and `normalized_gcd_monoid_of_gcd` noncomputably construct a `gcd_monoid`
  (resp. `normalized_gcd_monoid`) structure just from the `gcd` and its properties.

* `gcd_monoid_of_exists_gcd` and `normalized_gcd_monoid_of_exists_gcd` noncomputably construct a
  `gcd_monoid` (resp. `normalized_gcd_monoid`) structure just from a proof that any two elements
  have a (not necessarily normalized) `gcd`.

* `gcd_monoid_of_lcm` and `normalized_gcd_monoid_of_lcm` noncomputably construct a `gcd_monoid`
  (resp. `normalized_gcd_monoid`) structure just from the `lcm` and its properties.

* `gcd_monoid_of_exists_lcm` and `normalized_gcd_monoid_of_exists_lcm` noncomputably construct a
  `gcd_monoid` (resp. `normalized_gcd_monoid`) structure just from a proof that any two elements
  have a (not necessarily normalized) `lcm`.

## TODO

* Port GCD facts about nats, definition of coprime
* Generalize normalization monoids to commutative (cancellative) monoids with or without zero

## Tags

divisibility, gcd, lcm, normalize
-/

variable {α : Type _}

#print NormalizationMonoid /-
/-- Normalization monoid: multiplying with `norm_unit` gives a normal form for associated
elements. -/
@[protect_proj]
class NormalizationMonoid (α : Type _) [CancelCommMonoidWithZero α] where
  normUnit : α → αˣ
  normUnit_zero : norm_unit 0 = 1
  normUnit_mul : ∀ {a b}, a ≠ 0 → b ≠ 0 → norm_unit (a * b) = norm_unit a * norm_unit b
  normUnit_coe_units : ∀ u : αˣ, norm_unit u = u⁻¹
#align normalization_monoid NormalizationMonoid
-/

export NormalizationMonoid (normUnit normUnit_zero normUnit_mul normUnit_coe_units)

attribute [simp] norm_unit_coe_units norm_unit_zero norm_unit_mul

section NormalizationMonoid

variable [CancelCommMonoidWithZero α] [NormalizationMonoid α]

/- warning: norm_unit_one -> normUnit_one is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (NormalizationMonoid.normUnit.{u1} α _inst_1 _inst_2 (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (OfNat.ofNat.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (OfNat.mk.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (One.one.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulOneClass.toHasOne.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Units.mulOneClass.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (NormalizationMonoid.normUnit.{u1} α _inst_1 _inst_2 (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (OfNat.ofNat.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (One.toOfNat1.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (InvOneClass.toOne.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivInvOneMonoid.toInvOneClass.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivisionMonoid.toDivInvOneMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivisionCommMonoid.toDivisionMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (CommGroup.toDivisionCommMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Units.instCommGroupUnitsToMonoid.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))))
Case conversion may be inaccurate. Consider using '#align norm_unit_one normUnit_oneₓ'. -/
@[simp]
theorem normUnit_one : normUnit (1 : α) = 1 :=
  normUnit_coe_units 1
#align norm_unit_one normUnit_one

#print normalize /-
/-- Chooses an element of each associate class, by multiplying by `norm_unit` -/
def normalize : α →*₀ α where
  toFun x := x * normUnit x
  map_zero' := by simp
  map_one' := by rw [normUnit_one, Units.val_one, mul_one]
  map_mul' x y :=
    by_cases
      (fun hx : x = 0 => by
        rw [hx, MulZeroClass.zero_mul, MulZeroClass.zero_mul, MulZeroClass.zero_mul])
      fun hx =>
      by_cases
        (fun hy : y = 0 => by
          rw [hy, MulZeroClass.mul_zero, MulZeroClass.zero_mul, MulZeroClass.mul_zero])
        fun hy => by
        simp only [norm_unit_mul hx hy, Units.val_mul] <;> simp only [mul_assoc, mul_left_comm y]
#align normalize normalize
-/

/- warning: associated_normalize -> associated_normalize is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x)
Case conversion may be inaccurate. Consider using '#align associated_normalize associated_normalizeₓ'. -/
theorem associated_normalize (x : α) : Associated x (normalize x) :=
  ⟨_, rfl⟩
#align associated_normalize associated_normalize

/- warning: normalize_associated -> normalize_associated is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x) x
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Associated.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x) x
Case conversion may be inaccurate. Consider using '#align normalize_associated normalize_associatedₓ'. -/
theorem normalize_associated (x : α) : Associated (normalize x) x :=
  (associated_normalize _).symm
#align normalize_associated normalize_associated

/- warning: associated_normalize_iff -> associated_normalize_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α} {y : α}, Iff (Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) y)) (Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x y)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α} {y : α}, Iff (Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) y)) (Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x y)
Case conversion may be inaccurate. Consider using '#align associated_normalize_iff associated_normalize_iffₓ'. -/
theorem associated_normalize_iff {x y : α} : Associated x (normalize y) ↔ Associated x y :=
  ⟨fun h => h.trans (normalize_associated y), fun h => h.trans (associated_normalize y)⟩
#align associated_normalize_iff associated_normalize_iff

/- warning: normalize_associated_iff -> normalize_associated_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α} {y : α}, Iff (Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x) y) (Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x y)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α} {y : α}, Iff (Associated.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x) y) (Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x y)
Case conversion may be inaccurate. Consider using '#align normalize_associated_iff normalize_associated_iffₓ'. -/
theorem normalize_associated_iff {x y : α} : Associated (normalize x) y ↔ Associated x y :=
  ⟨fun h => (associated_normalize _).trans h, fun h => (normalize_associated _).trans h⟩
#align normalize_associated_iff normalize_associated_iff

/- warning: associates.mk_normalize -> Associates.mk_normalize is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Eq.{succ u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x)) (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Eq.{succ u1} (Associates.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1)))) (Associates.mk.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x)) (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x)
Case conversion may be inaccurate. Consider using '#align associates.mk_normalize Associates.mk_normalizeₓ'. -/
theorem Associates.mk_normalize (x : α) : Associates.mk (normalize x) = Associates.mk x :=
  Associates.mk_eq_mk_iff_associated.2 (normalize_associated _)
#align associates.mk_normalize Associates.mk_normalize

/- warning: normalize_apply -> normalize_apply is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) x ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (Units.hasCoe.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (NormalizationMonoid.normUnit.{u1} α _inst_1 _inst_2 x)))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) x (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (NormalizationMonoid.normUnit.{u1} α _inst_1 _inst_2 x)))
Case conversion may be inaccurate. Consider using '#align normalize_apply normalize_applyₓ'. -/
@[simp]
theorem normalize_apply (x : α) : normalize x = x * normUnit x :=
  rfl
#align normalize_apply normalize_apply

/- warning: normalize_zero -> normalize_zero is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (CommMonoidWithZero.toZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) _inst_1))))
Case conversion may be inaccurate. Consider using '#align normalize_zero normalize_zeroₓ'. -/
@[simp]
theorem normalize_zero : normalize (0 : α) = 0 :=
  normalize.map_zero
#align normalize_zero normalize_zero

/- warning: normalize_one -> normalize_one is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) 1 (One.toOfNat1.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (Monoid.toOne.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (MonoidWithZero.toMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) _inst_1))))))
Case conversion may be inaccurate. Consider using '#align normalize_one normalize_oneₓ'. -/
@[simp]
theorem normalize_one : normalize (1 : α) = 1 :=
  normalize.map_one
#align normalize_one normalize_one

/- warning: normalize_coe_units -> normalize_coe_units is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (u : Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (Units.hasCoe.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) u)) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (u : Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) 1 (One.toOfNat1.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) (Monoid.toOne.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) (MonoidWithZero.toMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) _inst_1))))))
Case conversion may be inaccurate. Consider using '#align normalize_coe_units normalize_coe_unitsₓ'. -/
theorem normalize_coe_units (u : αˣ) : normalize (u : α) = 1 := by simp
#align normalize_coe_units normalize_coe_units

/- warning: normalize_eq_zero -> normalize_eq_zero is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α}, Iff (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (Eq.{succ u1} α x (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α}, Iff (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))))) (Eq.{succ u1} α x (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))
Case conversion may be inaccurate. Consider using '#align normalize_eq_zero normalize_eq_zeroₓ'. -/
theorem normalize_eq_zero {x : α} : normalize x = 0 ↔ x = 0 :=
  ⟨fun hx => (associated_zero_iff_eq_zero x).1 <| hx ▸ associated_normalize _, by
    rintro rfl <;> exact normalize_zero⟩
#align normalize_eq_zero normalize_eq_zero

/- warning: normalize_eq_one -> normalize_eq_one is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α}, Iff (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α}, Iff (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) 1 (One.toOfNat1.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (Monoid.toOne.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))))))) (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) x)
Case conversion may be inaccurate. Consider using '#align normalize_eq_one normalize_eq_oneₓ'. -/
theorem normalize_eq_one {x : α} : normalize x = 1 ↔ IsUnit x :=
  ⟨fun hx => isUnit_iff_exists_inv.2 ⟨_, hx⟩, fun ⟨u, hu⟩ => hu ▸ normalize_coe_units u⟩
#align normalize_eq_one normalize_eq_one

/- warning: norm_unit_mul_norm_unit -> normUnit_mul_normUnit is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (NormalizationMonoid.normUnit.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (Units.hasCoe.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (NormalizationMonoid.normUnit.{u1} α _inst_1 _inst_2 a)))) (OfNat.ofNat.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (OfNat.mk.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (One.one.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulOneClass.toHasOne.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Units.mulOneClass.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (NormalizationMonoid.normUnit.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (NormalizationMonoid.normUnit.{u1} α _inst_1 _inst_2 a)))) (OfNat.ofNat.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (One.toOfNat1.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (InvOneClass.toOne.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivInvOneMonoid.toInvOneClass.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivisionMonoid.toDivInvOneMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivisionCommMonoid.toDivisionMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (CommGroup.toDivisionCommMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Units.instCommGroupUnitsToMonoid.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))))
Case conversion may be inaccurate. Consider using '#align norm_unit_mul_norm_unit normUnit_mul_normUnitₓ'. -/
@[simp]
theorem normUnit_mul_normUnit (a : α) : normUnit (a * normUnit a) = 1 :=
  by
  nontriviality α using Subsingleton.elim a 0
  obtain rfl | h := eq_or_ne a 0
  · rw [norm_unit_zero, MulZeroClass.zero_mul, norm_unit_zero]
  · rw [norm_unit_mul h (Units.ne_zero _), norm_unit_coe_units, mul_inv_eq_one]
#align norm_unit_mul_norm_unit normUnit_mul_normUnit

/- warning: normalize_idem -> normalize_idem is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (x : α), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x)) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x)
but is expected to have type
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(MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1)))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (fun (_x : (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1)))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MulOneClass.toMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MulZeroOneClass.toMulOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))))) (MulOneClass.toMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MulZeroOneClass.toMulOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1)))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MulZeroOneClass.toMulOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1)))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun 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(MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x)
Case conversion may be inaccurate. Consider using '#align normalize_idem normalize_idemₓ'. -/
theorem normalize_idem (x : α) : normalize (normalize x) = normalize x := by simp
#align normalize_idem normalize_idem

/- warning: normalize_eq_normalize -> normalize_eq_normalize is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {a : α} {b : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a) -> (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) a) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) b))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {a : α} {b : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) b))
Case conversion may be inaccurate. Consider using '#align normalize_eq_normalize normalize_eq_normalizeₓ'. -/
theorem normalize_eq_normalize {a b : α} (hab : a ∣ b) (hba : b ∣ a) : normalize a = normalize b :=
  by
  nontriviality α
  rcases associated_of_dvd_dvd hab hba with ⟨u, rfl⟩
  refine' by_cases (by rintro rfl <;> simp only [MulZeroClass.zero_mul]) fun ha : a ≠ 0 => _
  suffices a * ↑(norm_unit a) = a * ↑u * ↑(norm_unit a) * ↑u⁻¹ by
    simpa only [normalize_apply, mul_assoc, norm_unit_mul ha u.ne_zero, norm_unit_coe_units]
  calc
    a * ↑(norm_unit a) = a * ↑(norm_unit a) * ↑u * ↑u⁻¹ := (Units.mul_inv_cancel_right _ _).symm
    _ = a * ↑u * ↑(norm_unit a) * ↑u⁻¹ := by rw [mul_right_comm a]

#align normalize_eq_normalize normalize_eq_normalize

/- warning: normalize_eq_normalize_iff -> normalize_eq_normalize_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α} {y : α}, Iff (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) x) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) y)) (And (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) x y) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) y x))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {x : α} {y : α}, Iff (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) x) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) y)) (And (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) x y) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) y x))
Case conversion may be inaccurate. Consider using '#align normalize_eq_normalize_iff normalize_eq_normalize_iffₓ'. -/
theorem normalize_eq_normalize_iff {x y : α} : normalize x = normalize y ↔ x ∣ y ∧ y ∣ x :=
  ⟨fun h => ⟨Units.dvd_mul_right.1 ⟨_, h.symm⟩, Units.dvd_mul_right.1 ⟨_, h⟩⟩, fun ⟨hxy, hyx⟩ =>
    normalize_eq_normalize hxy hyx⟩
#align normalize_eq_normalize_iff normalize_eq_normalize_iff

/- warning: dvd_antisymm_of_normalize_eq -> dvd_antisymm_of_normalize_eq is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {a : α} {b : α}, (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) a) a) -> (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) b) b) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a) -> (Eq.{succ u1} α a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {a : α} {b : α}, (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) a) a) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) b) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) b) b) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a) -> (Eq.{succ u1} α a b)
Case conversion may be inaccurate. Consider using '#align dvd_antisymm_of_normalize_eq dvd_antisymm_of_normalize_eqₓ'. -/
theorem dvd_antisymm_of_normalize_eq {a b : α} (ha : normalize a = a) (hb : normalize b = b)
    (hab : a ∣ b) (hba : b ∣ a) : a = b :=
  ha ▸ hb ▸ normalize_eq_normalize hab hba
#align dvd_antisymm_of_normalize_eq dvd_antisymm_of_normalize_eq

/- warning: dvd_normalize_iff -> dvd_normalize_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {a : α} {b : α}, Iff (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) b)) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {a : α} {b : α}, Iff (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) b)) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b)
Case conversion may be inaccurate. Consider using '#align dvd_normalize_iff dvd_normalize_iffₓ'. -/
--can be proven by simp
theorem dvd_normalize_iff {a b : α} : a ∣ normalize b ↔ a ∣ b :=
  Units.dvd_mul_right
#align dvd_normalize_iff dvd_normalize_iff

/- warning: normalize_dvd_iff -> normalize_dvd_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {a : α} {b : α}, Iff (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) a) b) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] {a : α} {b : α}, Iff (Dvd.dvd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (semigroupDvd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (SemigroupWithZero.toSemigroup.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (MonoidWithZero.toSemigroupWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) _inst_1))))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) a) b) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b)
Case conversion may be inaccurate. Consider using '#align normalize_dvd_iff normalize_dvd_iffₓ'. -/
--can be proven by simp
theorem normalize_dvd_iff {a b : α} : normalize a ∣ b ↔ a ∣ b :=
  Units.mul_right_dvd
#align normalize_dvd_iff normalize_dvd_iff

end NormalizationMonoid

namespace Associates

variable [CancelCommMonoidWithZero α] [NormalizationMonoid α]

attribute [local instance] Associated.setoid

#print Associates.out /-
/-- Maps an element of `associates` back to the normalized element of its associate class -/
protected def out : Associates α → α :=
  Quotient.lift (normalize : α → α) fun a b ⟨u, hu⟩ =>
    hu ▸ normalize_eq_normalize ⟨_, rfl⟩ (Units.mul_right_dvd.2 <| dvd_refl a)
#align associates.out Associates.out
-/

/- warning: associates.out_mk -> Associates.out_mk is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (Associates.out.{u1} α _inst_1 _inst_2 (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) a)) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (Associates.out.{u1} α _inst_1 _inst_2 (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) a)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) a)
Case conversion may be inaccurate. Consider using '#align associates.out_mk Associates.out_mkₓ'. -/
@[simp]
theorem out_mk (a : α) : (Associates.mk a).out = normalize a :=
  rfl
#align associates.out_mk Associates.out_mk

/- warning: associates.out_one -> Associates.out_one is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} α (Associates.out.{u1} α _inst_1 _inst_2 (OfNat.ofNat.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (OfNat.mk.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (One.one.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.hasOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} α (Associates.out.{u1} α _inst_1 _inst_2 (OfNat.ofNat.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (One.toOfNat1.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instOneAssociates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))
Case conversion may be inaccurate. Consider using '#align associates.out_one Associates.out_oneₓ'. -/
@[simp]
theorem out_one : (1 : Associates α).out = 1 :=
  normalize_one
#align associates.out_one Associates.out_one

/- warning: associates.out_mul -> Associates.out_mul is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (b : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Eq.{succ u1} α (Associates.out.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (instHMul.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.hasMul.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) a b)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (Associates.out.{u1} α _inst_1 _inst_2 a) (Associates.out.{u1} α _inst_1 _inst_2 b))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (b : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Eq.{succ u1} α (Associates.out.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (instHMul.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instMulAssociatesToMonoid.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) a b)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (Associates.out.{u1} α _inst_1 _inst_2 a) (Associates.out.{u1} α _inst_1 _inst_2 b))
Case conversion may be inaccurate. Consider using '#align associates.out_mul Associates.out_mulₓ'. -/
theorem out_mul (a b : Associates α) : (a * b).out = a.out * b.out :=
  Quotient.induction_on₂ a b fun a b => by
    simp only [Associates.quotient_mk_eq_mk, out_mk, mk_mul_mk, normalize.map_mul]
#align associates.out_mul Associates.out_mul

/- warning: associates.dvd_out_iff -> Associates.dvd_out_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : α) (b : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Iff (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (Associates.out.{u1} α _inst_1 _inst_2 b)) (LE.le.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.preorder.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) a) b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : α) (b : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Iff (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (Associates.out.{u1} α _inst_1 _inst_2 b)) (LE.le.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instPreorderAssociatesToMonoid.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) a) b)
Case conversion may be inaccurate. Consider using '#align associates.dvd_out_iff Associates.dvd_out_iffₓ'. -/
theorem dvd_out_iff (a : α) (b : Associates α) : a ∣ b.out ↔ Associates.mk a ≤ b :=
  Quotient.inductionOn b <| by
    simp [Associates.out_mk, Associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff]
#align associates.dvd_out_iff Associates.dvd_out_iff

/- warning: associates.out_dvd_iff -> Associates.out_dvd_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : α) (b : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Iff (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (Associates.out.{u1} α _inst_1 _inst_2 b) a) (LE.le.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.preorder.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) b (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) a))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : α) (b : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Iff (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (Associates.out.{u1} α _inst_1 _inst_2 b) a) (LE.le.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instPreorderAssociatesToMonoid.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) b (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) a))
Case conversion may be inaccurate. Consider using '#align associates.out_dvd_iff Associates.out_dvd_iffₓ'. -/
theorem out_dvd_iff (a : α) (b : Associates α) : b.out ∣ a ↔ b ≤ Associates.mk a :=
  Quotient.inductionOn b <| by
    simp [Associates.out_mk, Associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff]
#align associates.out_dvd_iff Associates.out_dvd_iff

/- warning: associates.out_top -> Associates.out_top is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} α (Associates.out.{u1} α _inst_1 _inst_2 (Top.top.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.hasTop.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1], Eq.{succ u1} α (Associates.out.{u1} α _inst_1 _inst_2 (Top.top.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instTopAssociates.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)) (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))
Case conversion may be inaccurate. Consider using '#align associates.out_top Associates.out_topₓ'. -/
@[simp]
theorem out_top : (⊤ : Associates α).out = 0 :=
  normalize_zero
#align associates.out_top Associates.out_top

/- warning: associates.normalize_out -> Associates.normalize_out is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) (Associates.out.{u1} α _inst_1 _inst_2 a)) (Associates.out.{u1} α _inst_1 _inst_2 a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] (a : Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (Associates.out.{u1} α _inst_1 _inst_2 a)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) (Associates.out.{u1} α _inst_1 _inst_2 a)) (Associates.out.{u1} α _inst_1 _inst_2 a)
Case conversion may be inaccurate. Consider using '#align associates.normalize_out Associates.normalize_outₓ'. -/
@[simp]
theorem normalize_out (a : Associates α) : normalize a.out = a.out :=
  Quotient.inductionOn a normalize_idem
#align associates.normalize_out Associates.normalize_out

#print Associates.mk_out /-
@[simp]
theorem mk_out (a : Associates α) : Associates.mk a.out = a :=
  Quotient.inductionOn a mk_normalize
#align associates.mk_out Associates.mk_out
-/

#print Associates.out_injective /-
theorem out_injective : Function.Injective (Associates.out : _ → α) :=
  Function.LeftInverse.injective mk_out
#align associates.out_injective Associates.out_injective
-/

end Associates

#print GCDMonoid /-
/-- GCD monoid: a `cancel_comm_monoid_with_zero` with `gcd` (greatest common divisor) and
`lcm` (least common multiple) operations, determined up to a unit. The type class focuses on `gcd`
and we derive the corresponding `lcm` facts from `gcd`.
-/
@[protect_proj]
class GCDMonoid (α : Type _) [CancelCommMonoidWithZero α] where
  gcd : α → α → α
  lcm : α → α → α
  gcd_dvd_left : ∀ a b, gcd a b ∣ a
  gcd_dvd_right : ∀ a b, gcd a b ∣ b
  dvd_gcd : ∀ {a b c}, a ∣ c → a ∣ b → a ∣ gcd c b
  gcd_mul_lcm : ∀ a b, Associated (gcd a b * lcm a b) (a * b)
  lcm_zero_left : ∀ a, lcm 0 a = 0
  lcm_zero_right : ∀ a, lcm a 0 = 0
#align gcd_monoid GCDMonoid
-/

#print NormalizedGCDMonoid /-
/-- Normalized GCD monoid: a `cancel_comm_monoid_with_zero` with normalization and `gcd`
(greatest common divisor) and `lcm` (least common multiple) operations. In this setting `gcd` and
`lcm` form a bounded lattice on the associated elements where `gcd` is the infimum, `lcm` is the
supremum, `1` is bottom, and `0` is top. The type class focuses on `gcd` and we derive the
corresponding `lcm` facts from `gcd`.
-/
class NormalizedGCDMonoid (α : Type _) [CancelCommMonoidWithZero α] extends NormalizationMonoid α,
  GCDMonoid α where
  normalize_gcd : ∀ a b, normalize (gcd a b) = gcd a b
  normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b
#align normalized_gcd_monoid NormalizedGCDMonoid
-/

export GCDMonoid (gcd lcm gcd_dvd_left gcd_dvd_right dvd_gcd lcm_zero_left lcm_zero_right)

attribute [simp] lcm_zero_left lcm_zero_right

section GCDMonoid

variable [CancelCommMonoidWithZero α]

/- warning: normalize_gcd -> normalize_gcd is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b)) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b)) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b)
Case conversion may be inaccurate. Consider using '#align normalize_gcd normalize_gcdₓ'. -/
@[simp]
theorem normalize_gcd [NormalizedGCDMonoid α] : ∀ a b : α, normalize (gcd a b) = gcd a b :=
  NormalizedGCDMonoid.normalize_gcd
#align normalize_gcd normalize_gcd

/- warning: gcd_mul_lcm -> gcd_mul_lcm is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 a b)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 a b)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b)
Case conversion may be inaccurate. Consider using '#align gcd_mul_lcm gcd_mul_lcmₓ'. -/
theorem gcd_mul_lcm [GCDMonoid α] : ∀ a b : α, Associated (gcd a b * lcm a b) (a * b) :=
  GCDMonoid.gcd_mul_lcm
#align gcd_mul_lcm gcd_mul_lcm

section Gcd

#print dvd_gcd_iff /-
theorem dvd_gcd_iff [GCDMonoid α] (a b c : α) : a ∣ gcd b c ↔ a ∣ b ∧ a ∣ c :=
  Iff.intro (fun h => ⟨h.trans (gcd_dvd_left _ _), h.trans (gcd_dvd_right _ _)⟩) fun ⟨hab, hac⟩ =>
    dvd_gcd hab hac
#align dvd_gcd_iff dvd_gcd_iff
-/

#print gcd_comm /-
theorem gcd_comm [NormalizedGCDMonoid α] (a b : α) : gcd a b = gcd b a :=
  dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)
    (dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _))
    (dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _))
#align gcd_comm gcd_comm
-/

#print gcd_comm' /-
theorem gcd_comm' [GCDMonoid α] (a b : α) : Associated (gcd a b) (gcd b a) :=
  associated_of_dvd_dvd (dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _))
    (dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _))
#align gcd_comm' gcd_comm'
-/

#print gcd_assoc /-
theorem gcd_assoc [NormalizedGCDMonoid α] (m n k : α) : gcd (gcd m n) k = gcd m (gcd n k) :=
  dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)
    (dvd_gcd ((gcd_dvd_left (gcd m n) k).trans (gcd_dvd_left m n))
      (dvd_gcd ((gcd_dvd_left (gcd m n) k).trans (gcd_dvd_right m n)) (gcd_dvd_right (gcd m n) k)))
    (dvd_gcd
      (dvd_gcd (gcd_dvd_left m (gcd n k)) ((gcd_dvd_right m (gcd n k)).trans (gcd_dvd_left n k)))
      ((gcd_dvd_right m (gcd n k)).trans (gcd_dvd_right n k)))
#align gcd_assoc gcd_assoc
-/

#print gcd_assoc' /-
theorem gcd_assoc' [GCDMonoid α] (m n k : α) : Associated (gcd (gcd m n) k) (gcd m (gcd n k)) :=
  associated_of_dvd_dvd
    (dvd_gcd ((gcd_dvd_left (gcd m n) k).trans (gcd_dvd_left m n))
      (dvd_gcd ((gcd_dvd_left (gcd m n) k).trans (gcd_dvd_right m n)) (gcd_dvd_right (gcd m n) k)))
    (dvd_gcd
      (dvd_gcd (gcd_dvd_left m (gcd n k)) ((gcd_dvd_right m (gcd n k)).trans (gcd_dvd_left n k)))
      ((gcd_dvd_right m (gcd n k)).trans (gcd_dvd_right n k)))
#align gcd_assoc' gcd_assoc'
-/

instance [NormalizedGCDMonoid α] : IsCommutative α gcd :=
  ⟨gcd_comm⟩

instance [NormalizedGCDMonoid α] : IsAssociative α gcd :=
  ⟨gcd_assoc⟩

/- warning: gcd_eq_normalize -> gcd_eq_normalize is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] {a : α} {b : α} {c : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) c) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b)) -> (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) c))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] {a : α} {b : α} {c : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) c) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b)) -> (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) c))
Case conversion may be inaccurate. Consider using '#align gcd_eq_normalize gcd_eq_normalizeₓ'. -/
theorem gcd_eq_normalize [NormalizedGCDMonoid α] {a b c : α} (habc : gcd a b ∣ c)
    (hcab : c ∣ gcd a b) : gcd a b = normalize c :=
  normalize_gcd a b ▸ normalize_eq_normalize habc hcab
#align gcd_eq_normalize gcd_eq_normalize

/- warning: gcd_zero_left -> gcd_zero_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))))) a) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
Case conversion may be inaccurate. Consider using '#align gcd_zero_left gcd_zero_leftₓ'. -/
@[simp]
theorem gcd_zero_left [NormalizedGCDMonoid α] (a : α) : gcd 0 a = normalize a :=
  gcd_eq_normalize (gcd_dvd_right 0 a) (dvd_gcd (dvd_zero _) (dvd_refl a))
#align gcd_zero_left gcd_zero_left

/- warning: gcd_zero_left' -> gcd_zero_left' is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))))) a) a
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) a) a
Case conversion may be inaccurate. Consider using '#align gcd_zero_left' gcd_zero_left'ₓ'. -/
theorem gcd_zero_left' [GCDMonoid α] (a : α) : Associated (gcd 0 a) a :=
  associated_of_dvd_dvd (gcd_dvd_right 0 a) (dvd_gcd (dvd_zero _) (dvd_refl a))
#align gcd_zero_left' gcd_zero_left'

/- warning: gcd_zero_right -> gcd_zero_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
Case conversion may be inaccurate. Consider using '#align gcd_zero_right gcd_zero_rightₓ'. -/
@[simp]
theorem gcd_zero_right [NormalizedGCDMonoid α] (a : α) : gcd a 0 = normalize a :=
  gcd_eq_normalize (gcd_dvd_left a 0) (dvd_gcd (dvd_refl a) (dvd_zero _))
#align gcd_zero_right gcd_zero_right

/- warning: gcd_zero_right' -> gcd_zero_right' is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) a
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a
Case conversion may be inaccurate. Consider using '#align gcd_zero_right' gcd_zero_right'ₓ'. -/
theorem gcd_zero_right' [GCDMonoid α] (a : α) : Associated (gcd a 0) a :=
  associated_of_dvd_dvd (gcd_dvd_left a 0) (dvd_gcd (dvd_refl a) (dvd_zero _))
#align gcd_zero_right' gcd_zero_right'

/- warning: gcd_eq_zero_iff -> gcd_eq_zero_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α), Iff (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (And (Eq.{succ u1} α a (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (Eq.{succ u1} α b (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α), Iff (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (And (Eq.{succ u1} α a (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (Eq.{succ u1} α b (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))
Case conversion may be inaccurate. Consider using '#align gcd_eq_zero_iff gcd_eq_zero_iffₓ'. -/
@[simp]
theorem gcd_eq_zero_iff [GCDMonoid α] (a b : α) : gcd a b = 0 ↔ a = 0 ∧ b = 0 :=
  Iff.intro
    (fun h => by
      let ⟨ca, ha⟩ := gcd_dvd_left a b
      let ⟨cb, hb⟩ := gcd_dvd_right a b
      rw [h, MulZeroClass.zero_mul] at ha hb <;> exact ⟨ha, hb⟩)
    fun ⟨ha, hb⟩ => by
    rw [ha, hb, ← zero_dvd_iff]
    apply dvd_gcd <;> rfl
#align gcd_eq_zero_iff gcd_eq_zero_iff

/- warning: gcd_one_left -> gcd_one_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))))) a) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))
Case conversion may be inaccurate. Consider using '#align gcd_one_left gcd_one_leftₓ'. -/
@[simp]
theorem gcd_one_left [NormalizedGCDMonoid α] (a : α) : gcd 1 a = 1 :=
  dvd_antisymm_of_normalize_eq (normalize_gcd _ _) normalize_one (gcd_dvd_left _ _) (one_dvd _)
#align gcd_one_left gcd_one_left

/- warning: gcd_one_left' -> gcd_one_left' is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))))) a) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))
Case conversion may be inaccurate. Consider using '#align gcd_one_left' gcd_one_left'ₓ'. -/
@[simp]
theorem gcd_one_left' [GCDMonoid α] (a : α) : Associated (gcd 1 a) 1 :=
  associated_of_dvd_dvd (gcd_dvd_left _ _) (one_dvd _)
#align gcd_one_left' gcd_one_left'

/- warning: gcd_one_right -> gcd_one_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))
Case conversion may be inaccurate. Consider using '#align gcd_one_right gcd_one_rightₓ'. -/
@[simp]
theorem gcd_one_right [NormalizedGCDMonoid α] (a : α) : gcd a 1 = 1 :=
  dvd_antisymm_of_normalize_eq (normalize_gcd _ _) normalize_one (gcd_dvd_right _ _) (one_dvd _)
#align gcd_one_right gcd_one_right

/- warning: gcd_one_right' -> gcd_one_right' is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))
Case conversion may be inaccurate. Consider using '#align gcd_one_right' gcd_one_right'ₓ'. -/
@[simp]
theorem gcd_one_right' [GCDMonoid α] (a : α) : Associated (gcd a 1) 1 :=
  associated_of_dvd_dvd (gcd_dvd_right _ _) (one_dvd _)
#align gcd_one_right' gcd_one_right'

#print gcd_dvd_gcd /-
theorem gcd_dvd_gcd [GCDMonoid α] {a b c d : α} (hab : a ∣ b) (hcd : c ∣ d) : gcd a c ∣ gcd b d :=
  dvd_gcd ((gcd_dvd_left _ _).trans hab) ((gcd_dvd_right _ _).trans hcd)
#align gcd_dvd_gcd gcd_dvd_gcd
-/

/- warning: gcd_same -> gcd_same is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a a) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
Case conversion may be inaccurate. Consider using '#align gcd_same gcd_sameₓ'. -/
@[simp]
theorem gcd_same [NormalizedGCDMonoid α] (a : α) : gcd a a = normalize a :=
  gcd_eq_normalize (gcd_dvd_left _ _) (dvd_gcd (dvd_refl a) (dvd_refl a))
#align gcd_same gcd_same

/- warning: gcd_mul_left -> gcd_mul_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a c)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) b c))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a c)) (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) α ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (MulZeroClass.toMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (MulZeroOneClass.toMulZeroClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) _inst_1)))))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) b c))
Case conversion may be inaccurate. Consider using '#align gcd_mul_left gcd_mul_leftₓ'. -/
@[simp]
theorem gcd_mul_left [NormalizedGCDMonoid α] (a b c : α) :
    gcd (a * b) (a * c) = normalize a * gcd b c :=
  by_cases (by rintro rfl <;> simp only [MulZeroClass.zero_mul, gcd_zero_left, normalize_zero])
    fun ha : a ≠ 0 =>
    suffices gcd (a * b) (a * c) = normalize (a * gcd b c) by
      simpa only [normalize.map_mul, normalize_gcd]
    let ⟨d, Eq⟩ := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c)
    gcd_eq_normalize
      (Eq.symm ▸ mul_dvd_mul_left a <|
        show d ∣ gcd b c from
          dvd_gcd ((mul_dvd_mul_iff_left ha).1 <| Eq ▸ gcd_dvd_left _ _)
            ((mul_dvd_mul_iff_left ha).1 <| Eq ▸ gcd_dvd_right _ _))
      (dvd_gcd (mul_dvd_mul_left a <| gcd_dvd_left _ _) (mul_dvd_mul_left a <| gcd_dvd_right _ _))
#align gcd_mul_left gcd_mul_left

/- warning: gcd_mul_left' -> gcd_mul_left' is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a c)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 b c))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a c)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 b c))
Case conversion may be inaccurate. Consider using '#align gcd_mul_left' gcd_mul_left'ₓ'. -/
theorem gcd_mul_left' [GCDMonoid α] (a b c : α) : Associated (gcd (a * b) (a * c)) (a * gcd b c) :=
  by
  obtain rfl | ha := eq_or_ne a 0
  · simp only [MulZeroClass.zero_mul, gcd_zero_left']
  obtain ⟨d, eq⟩ := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c)
  apply associated_of_dvd_dvd
  · rw [Eq]
    apply mul_dvd_mul_left
    exact
      dvd_gcd ((mul_dvd_mul_iff_left ha).1 <| Eq ▸ gcd_dvd_left _ _)
        ((mul_dvd_mul_iff_left ha).1 <| Eq ▸ gcd_dvd_right _ _)
  · exact dvd_gcd (mul_dvd_mul_left a <| gcd_dvd_left _ _) (mul_dvd_mul_left a <| gcd_dvd_right _ _)
#align gcd_mul_left' gcd_mul_left'

/- warning: gcd_mul_right -> gcd_mul_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) b a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) c a)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) b c) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) b a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) c a)) (HMul.hMul.{u1, u1, u1} α ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) b c) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a))
Case conversion may be inaccurate. Consider using '#align gcd_mul_right gcd_mul_rightₓ'. -/
@[simp]
theorem gcd_mul_right [NormalizedGCDMonoid α] (a b c : α) :
    gcd (b * a) (c * a) = gcd b c * normalize a := by simp only [mul_comm, gcd_mul_left]
#align gcd_mul_right gcd_mul_right

/- warning: gcd_mul_right' -> gcd_mul_right' is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) b a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) c a)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 b c) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) b a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) c a)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 b c) a)
Case conversion may be inaccurate. Consider using '#align gcd_mul_right' gcd_mul_right'ₓ'. -/
@[simp]
theorem gcd_mul_right' [GCDMonoid α] (a b c : α) : Associated (gcd (b * a) (c * a)) (gcd b c * a) :=
  by simp only [mul_comm, gcd_mul_left']
#align gcd_mul_right' gcd_mul_right'

/- warning: gcd_eq_left_iff -> gcd_eq_left_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a) a) -> (Iff (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) a) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a) a) -> (Iff (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) a) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b))
Case conversion may be inaccurate. Consider using '#align gcd_eq_left_iff gcd_eq_left_iffₓ'. -/
theorem gcd_eq_left_iff [NormalizedGCDMonoid α] (a b : α) (h : normalize a = a) :
    gcd a b = a ↔ a ∣ b :=
  Iff.intro (fun eq => Eq ▸ gcd_dvd_right _ _) fun hab =>
    dvd_antisymm_of_normalize_eq (normalize_gcd _ _) h (gcd_dvd_left _ _) (dvd_gcd (dvd_refl a) hab)
#align gcd_eq_left_iff gcd_eq_left_iff

/- warning: gcd_eq_right_iff -> gcd_eq_right_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) b) b) -> (Iff (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) b) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) b) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) b) b) -> (Iff (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) b) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a))
Case conversion may be inaccurate. Consider using '#align gcd_eq_right_iff gcd_eq_right_iffₓ'. -/
theorem gcd_eq_right_iff [NormalizedGCDMonoid α] (a b : α) (h : normalize b = b) :
    gcd a b = b ↔ b ∣ a := by simpa only [gcd_comm a b] using gcd_eq_left_iff b a h
#align gcd_eq_right_iff gcd_eq_right_iff

/- warning: gcd_dvd_gcd_mul_left -> gcd_dvd_gcd_mul_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) k m) n)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) k m) n)
Case conversion may be inaccurate. Consider using '#align gcd_dvd_gcd_mul_left gcd_dvd_gcd_mul_leftₓ'. -/
theorem gcd_dvd_gcd_mul_left [GCDMonoid α] (m n k : α) : gcd m n ∣ gcd (k * m) n :=
  gcd_dvd_gcd (dvd_mul_left _ _) dvd_rfl
#align gcd_dvd_gcd_mul_left gcd_dvd_gcd_mul_left

/- warning: gcd_dvd_gcd_mul_right -> gcd_dvd_gcd_mul_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m k) n)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m k) n)
Case conversion may be inaccurate. Consider using '#align gcd_dvd_gcd_mul_right gcd_dvd_gcd_mul_rightₓ'. -/
theorem gcd_dvd_gcd_mul_right [GCDMonoid α] (m n k : α) : gcd m n ∣ gcd (m * k) n :=
  gcd_dvd_gcd (dvd_mul_right _ _) dvd_rfl
#align gcd_dvd_gcd_mul_right gcd_dvd_gcd_mul_right

/- warning: gcd_dvd_gcd_mul_left_right -> gcd_dvd_gcd_mul_left_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) k n))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) k n))
Case conversion may be inaccurate. Consider using '#align gcd_dvd_gcd_mul_left_right gcd_dvd_gcd_mul_left_rightₓ'. -/
theorem gcd_dvd_gcd_mul_left_right [GCDMonoid α] (m n k : α) : gcd m n ∣ gcd m (k * n) :=
  gcd_dvd_gcd dvd_rfl (dvd_mul_left _ _)
#align gcd_dvd_gcd_mul_left_right gcd_dvd_gcd_mul_left_right

/- warning: gcd_dvd_gcd_mul_right_right -> gcd_dvd_gcd_mul_right_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) n k))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 m (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) n k))
Case conversion may be inaccurate. Consider using '#align gcd_dvd_gcd_mul_right_right gcd_dvd_gcd_mul_right_rightₓ'. -/
theorem gcd_dvd_gcd_mul_right_right [GCDMonoid α] (m n k : α) : gcd m n ∣ gcd m (n * k) :=
  gcd_dvd_gcd dvd_rfl (dvd_mul_right _ _)
#align gcd_dvd_gcd_mul_right_right gcd_dvd_gcd_mul_right_right

#print Associated.gcd_eq_left /-
theorem Associated.gcd_eq_left [NormalizedGCDMonoid α] {m n : α} (h : Associated m n) (k : α) :
    gcd m k = gcd n k :=
  dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _) (gcd_dvd_gcd h.Dvd dvd_rfl)
    (gcd_dvd_gcd h.symm.Dvd dvd_rfl)
#align associated.gcd_eq_left Associated.gcd_eq_left
-/

#print Associated.gcd_eq_right /-
theorem Associated.gcd_eq_right [NormalizedGCDMonoid α] {m n : α} (h : Associated m n) (k : α) :
    gcd k m = gcd k n :=
  dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _) (gcd_dvd_gcd dvd_rfl h.Dvd)
    (gcd_dvd_gcd dvd_rfl h.symm.Dvd)
#align associated.gcd_eq_right Associated.gcd_eq_right
-/

/- warning: dvd_gcd_mul_of_dvd_mul -> dvd_gcd_mul_of_dvd_mul is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {m : α} {n : α} {k : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k m) n))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {m : α} {n : α} {k : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k m) n))
Case conversion may be inaccurate. Consider using '#align dvd_gcd_mul_of_dvd_mul dvd_gcd_mul_of_dvd_mulₓ'. -/
theorem dvd_gcd_mul_of_dvd_mul [GCDMonoid α] {m n k : α} (H : k ∣ m * n) : k ∣ gcd k m * n :=
  (dvd_gcd (dvd_mul_right _ n) H).trans (gcd_mul_right' n k m).Dvd
#align dvd_gcd_mul_of_dvd_mul dvd_gcd_mul_of_dvd_mul

/- warning: dvd_mul_gcd_of_dvd_mul -> dvd_mul_gcd_of_dvd_mul is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {m : α} {n : α} {k : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k n)))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {m : α} {n : α} {k : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k n)))
Case conversion may be inaccurate. Consider using '#align dvd_mul_gcd_of_dvd_mul dvd_mul_gcd_of_dvd_mulₓ'. -/
theorem dvd_mul_gcd_of_dvd_mul [GCDMonoid α] {m n k : α} (H : k ∣ m * n) : k ∣ m * gcd k n :=
  by
  rw [mul_comm] at H⊢
  exact dvd_gcd_mul_of_dvd_mul H
#align dvd_mul_gcd_of_dvd_mul dvd_mul_gcd_of_dvd_mul

/- warning: exists_dvd_and_dvd_of_dvd_mul -> exists_dvd_and_dvd_of_dvd_mul is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {m : α} {n : α} {k : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) -> (Exists.{succ u1} α (fun (d₁ : α) => Exists.{succ u1} α (fun (d₂ : α) => And (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ m) (And (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₂ n) (Eq.{succ u1} α k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) d₁ d₂))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {m : α} {n : α} {k : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) -> (Exists.{succ u1} α (fun (d₁ : α) => Exists.{succ u1} α (fun (d₂ : α) => And (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ m) (And (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₂ n) (Eq.{succ u1} α k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) d₁ d₂))))))
Case conversion may be inaccurate. Consider using '#align exists_dvd_and_dvd_of_dvd_mul exists_dvd_and_dvd_of_dvd_mulₓ'. -/
/-- Represent a divisor of `m * n` as a product of a divisor of `m` and a divisor of `n`.

In other words, the nonzero elements of a `gcd_monoid` form a decomposition monoid
(more widely known as a pre-Schreier domain in the context of rings).

Note: In general, this representation is highly non-unique.

See `nat.prod_dvd_and_dvd_of_dvd_prod` for a constructive version on `ℕ`.  -/
theorem exists_dvd_and_dvd_of_dvd_mul [GCDMonoid α] {m n k : α} (H : k ∣ m * n) :
    ∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂ :=
  by
  by_cases h0 : gcd k m = 0
  · rw [gcd_eq_zero_iff] at h0
    rcases h0 with ⟨rfl, rfl⟩
    refine' ⟨0, n, dvd_refl 0, dvd_refl n, _⟩
    simp
  · obtain ⟨a, ha⟩ := gcd_dvd_left k m
    refine' ⟨gcd k m, a, gcd_dvd_right _ _, _, ha⟩
    suffices h : gcd k m * a ∣ gcd k m * n
    · cases' h with b hb
      use b
      rw [mul_assoc] at hb
      apply mul_left_cancel₀ h0 hb
    rw [← ha]
    exact dvd_gcd_mul_of_dvd_mul H
#align exists_dvd_and_dvd_of_dvd_mul exists_dvd_and_dvd_of_dvd_mul

/- warning: dvd_mul -> dvd_mul is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {k : α} {m : α} {n : α}, Iff (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) (Exists.{succ u1} α (fun (d₁ : α) => Exists.{succ u1} α (fun (d₂ : α) => And (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ m) (And (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₂ n) (Eq.{succ u1} α k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) d₁ d₂))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {k : α} {m : α} {n : α}, Iff (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) (Exists.{succ u1} α (fun (d₁ : α) => Exists.{succ u1} α (fun (d₂ : α) => And (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ m) (And (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₂ n) (Eq.{succ u1} α k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) d₁ d₂))))))
Case conversion may be inaccurate. Consider using '#align dvd_mul dvd_mulₓ'. -/
theorem dvd_mul [GCDMonoid α] {k m n : α} : k ∣ m * n ↔ ∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂ :=
  by
  refine' ⟨exists_dvd_and_dvd_of_dvd_mul, _⟩
  rintro ⟨d₁, d₂, hy, hz, rfl⟩
  exact mul_dvd_mul hy hz
#align dvd_mul dvd_mul

/- warning: gcd_mul_dvd_mul_gcd -> gcd_mul_dvd_mul_gcd is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (k : α) (m : α) (n : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k m) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k n))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (k : α) (m : α) (n : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m n)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k m) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 k n))
Case conversion may be inaccurate. Consider using '#align gcd_mul_dvd_mul_gcd gcd_mul_dvd_mul_gcdₓ'. -/
theorem gcd_mul_dvd_mul_gcd [GCDMonoid α] (k m n : α) : gcd k (m * n) ∣ gcd k m * gcd k n :=
  by
  obtain ⟨m', n', hm', hn', h⟩ := exists_dvd_and_dvd_of_dvd_mul (gcd_dvd_right k (m * n))
  replace h : gcd k (m * n) = m' * n' := h
  rw [h]
  have hm'n' : m' * n' ∣ k := h ▸ gcd_dvd_left _ _
  apply mul_dvd_mul
  · have hm'k : m' ∣ k := (dvd_mul_right m' n').trans hm'n'
    exact dvd_gcd hm'k hm'
  · have hn'k : n' ∣ k := (dvd_mul_left n' m').trans hm'n'
    exact dvd_gcd hn'k hn'
#align gcd_mul_dvd_mul_gcd gcd_mul_dvd_mul_gcd

#print gcd_pow_right_dvd_pow_gcd /-
theorem gcd_pow_right_dvd_pow_gcd [GCDMonoid α] {a b : α} {k : ℕ} : gcd a (b ^ k) ∣ gcd a b ^ k :=
  by
  by_cases hg : gcd a b = 0
  · rw [gcd_eq_zero_iff] at hg
    rcases hg with ⟨rfl, rfl⟩
    exact
      (gcd_zero_left' (0 ^ k : α)).Dvd.trans
        (pow_dvd_pow_of_dvd (gcd_zero_left' (0 : α)).symm.Dvd _)
  · induction' k with k hk
    · simp only [pow_zero]
      exact (gcd_one_right' a).Dvd
    rw [pow_succ, pow_succ]
    trans gcd a b * gcd a (b ^ k)
    apply gcd_mul_dvd_mul_gcd a b (b ^ k)
    exact (mul_dvd_mul_iff_left hg).mpr hk
#align gcd_pow_right_dvd_pow_gcd gcd_pow_right_dvd_pow_gcd
-/

#print gcd_pow_left_dvd_pow_gcd /-
theorem gcd_pow_left_dvd_pow_gcd [GCDMonoid α] {a b : α} {k : ℕ} : gcd (a ^ k) b ∣ gcd a b ^ k :=
  calc
    gcd (a ^ k) b ∣ gcd b (a ^ k) := (gcd_comm' _ _).Dvd
    _ ∣ gcd b a ^ k := gcd_pow_right_dvd_pow_gcd
    _ ∣ gcd a b ^ k := pow_dvd_pow_of_dvd (gcd_comm' _ _).Dvd _

#align gcd_pow_left_dvd_pow_gcd gcd_pow_left_dvd_pow_gcd
-/

/- warning: pow_dvd_of_mul_eq_pow -> pow_dvd_of_mul_eq_pow is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {a : α} {b : α} {c : α} {d₁ : α} {d₂ : α}, (Ne.{succ u1} α a (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) -> (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b)) -> (forall {k : Nat}, (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c k)) -> (Eq.{succ u1} α c (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) d₁ d₂)) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ a) -> (And (Ne.{succ u1} α (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ k) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ k) a)))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {a : α} {b : α} {c : α} {d₁ : α} {d₂ : α}, (Ne.{succ u1} α a (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) -> (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b)) -> (forall {k : Nat}, (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c k)) -> (Eq.{succ u1} α c (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) d₁ d₂)) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ a) -> (And (Ne.{succ u1} α (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ k) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d₁ k) a)))
Case conversion may be inaccurate. Consider using '#align pow_dvd_of_mul_eq_pow pow_dvd_of_mul_eq_powₓ'. -/
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a :=
  by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  have h3 : d₁ ^ k ∣ a := by
    apply (dvd_gcd_mul_of_dvd_mul h2).trans
    rw [IsUnit.mul_left_dvd _ _ _ h1]
  have h4 : d₁ ^ k ≠ 0 := by
    intro hdk
    rw [hdk] at h3
    apply absurd (zero_dvd_iff.mp h3) ha
  exact ⟨h4, h3⟩
#align pow_dvd_of_mul_eq_pow pow_dvd_of_mul_eq_pow

/- warning: exists_associated_pow_of_mul_eq_pow -> exists_associated_pow_of_mul_eq_pow is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b)) -> (forall {k : Nat}, (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c k)) -> (Exists.{succ u1} α (fun (d : α) => Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d k) a)))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b)) -> (forall {k : Nat}, (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c k)) -> (Exists.{succ u1} α (fun (d : α) => Associated.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d k) a)))
Case conversion may be inaccurate. Consider using '#align exists_associated_pow_of_mul_eq_pow exists_associated_pow_of_mul_eq_powₓ'. -/
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a :=
  by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      rw [ha, MulZeroClass.zero_mul, pow_zero]
      apply zero_ne_one
    · rw [zero_pow hk]
  by_cases hb : b = 0
  · use 1
    rw [one_pow]
    apply (associated_one_iff_is_unit.mpr hab).symm.trans
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  rw [mul_comm] at h hc
  rw [(gcd_comm' a b).isUnit_iff] at hab
  obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂
  rw [ha', hb', hc, mul_pow] at h
  have h' : a' * b' = 1 := by
    apply (mul_right_inj' h0₁).mp
    rw [mul_one]
    apply (mul_right_inj' h0₂).mp
    rw [← h]
    rw [mul_assoc, mul_comm a', ← mul_assoc _ b', ← mul_assoc b', mul_comm b']
  use Units.mkOfMulEqOne _ _ h'
  rw [Units.val_mkOfMulEqOne, ha']
#align exists_associated_pow_of_mul_eq_pow exists_associated_pow_of_mul_eq_pow

/- warning: exists_eq_pow_of_mul_eq_pow -> exists_eq_pow_of_mul_eq_pow is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] [_inst_3 : Unique.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b)) -> (forall {k : Nat}, (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c k)) -> (Exists.{succ u1} α (fun (d : α) => Eq.{succ u1} α a (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d k))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] [_inst_3 : Unique.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (GCDMonoid.gcd.{u1} α _inst_1 _inst_2 a b)) -> (forall {k : Nat}, (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c k)) -> (Exists.{succ u1} α (fun (d : α) => Eq.{succ u1} α a (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) d k))))
Case conversion may be inaccurate. Consider using '#align exists_eq_pow_of_mul_eq_pow exists_eq_pow_of_mul_eq_powₓ'. -/
theorem exists_eq_pow_of_mul_eq_pow [GCDMonoid α] [Unique αˣ] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, a = d ^ k :=
  let ⟨d, hd⟩ := exists_associated_pow_of_mul_eq_pow hab h
  ⟨d, (associated_iff_eq.mp hd).symm⟩
#align exists_eq_pow_of_mul_eq_pow exists_eq_pow_of_mul_eq_pow

/- warning: gcd_greatest -> gcd_greatest is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_2 : CancelCommMonoidWithZero.{u1} α] [_inst_3 : NormalizedGCDMonoid.{u1} α _inst_2] {a : α} {b : α} {d : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) d a) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) d b) -> (forall (e : α), (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) e a) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) e b) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) e d)) -> (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_2 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_2 _inst_3) a b) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) (normalize.{u1} α _inst_2 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_2 _inst_3)) d))
but is expected to have type
  forall {α : Type.{u1}} [_inst_2 : CancelCommMonoidWithZero.{u1} α] [_inst_3 : NormalizedGCDMonoid.{u1} α _inst_2] {a : α} {b : α} {d : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) d a) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) d b) -> (forall (e : α), (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) e a) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) e b) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) e d)) -> (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_2 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_2 _inst_3) a b) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))))) (normalize.{u1} α _inst_2 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_2 _inst_3)) d))
Case conversion may be inaccurate. Consider using '#align gcd_greatest gcd_greatestₓ'. -/
theorem gcd_greatest {α : Type _} [CancelCommMonoidWithZero α] [NormalizedGCDMonoid α] {a b d : α}
    (hda : d ∣ a) (hdb : d ∣ b) (hd : ∀ e : α, e ∣ a → e ∣ b → e ∣ d) :
    GCDMonoid.gcd a b = normalize d :=
  haveI h := hd _ (GCDMonoid.gcd_dvd_left a b) (GCDMonoid.gcd_dvd_right a b)
  gcd_eq_normalize h (GCDMonoid.dvd_gcd hda hdb)
#align gcd_greatest gcd_greatest

#print gcd_greatest_associated /-
theorem gcd_greatest_associated {α : Type _} [CancelCommMonoidWithZero α] [GCDMonoid α] {a b d : α}
    (hda : d ∣ a) (hdb : d ∣ b) (hd : ∀ e : α, e ∣ a → e ∣ b → e ∣ d) :
    Associated d (GCDMonoid.gcd a b) :=
  haveI h := hd _ (GCDMonoid.gcd_dvd_left a b) (GCDMonoid.gcd_dvd_right a b)
  associated_of_dvd_dvd (GCDMonoid.dvd_gcd hda hdb) h
#align gcd_greatest_associated gcd_greatest_associated
-/

/- warning: is_unit_gcd_of_eq_mul_gcd -> isUnit_gcd_of_eq_mul_gcd is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_2 : CancelCommMonoidWithZero.{u1} α] [_inst_3 : GCDMonoid.{u1} α _inst_2] {x : α} {y : α} {x' : α} {y' : α}, (Eq.{succ u1} α x (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) x')) -> (Eq.{succ u1} α y (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) y')) -> (Ne.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))))))) -> (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x' y'))
but is expected to have type
  forall {α : Type.{u1}} [_inst_2 : CancelCommMonoidWithZero.{u1} α] [_inst_3 : GCDMonoid.{u1} α _inst_2] {x : α} {y : α} {x' : α} {y' : α}, (Eq.{succ u1} α x (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) x')) -> (Eq.{succ u1} α y (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) y')) -> (Ne.{succ u1} α (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))))) -> (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x' y'))
Case conversion may be inaccurate. Consider using '#align is_unit_gcd_of_eq_mul_gcd isUnit_gcd_of_eq_mul_gcdₓ'. -/
theorem isUnit_gcd_of_eq_mul_gcd {α : Type _} [CancelCommMonoidWithZero α] [GCDMonoid α]
    {x y x' y' : α} (ex : x = gcd x y * x') (ey : y = gcd x y * y') (h : gcd x y ≠ 0) :
    IsUnit (gcd x' y') := by
  rw [← associated_one_iff_isUnit]
  refine' Associated.of_mul_left _ (Associated.refl <| gcd x y) h
  convert(gcd_mul_left' _ _ _).symm using 1
  rw [← ex, ← ey, mul_one]
#align is_unit_gcd_of_eq_mul_gcd isUnit_gcd_of_eq_mul_gcd

/- warning: extract_gcd -> extract_gcd is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_2 : CancelCommMonoidWithZero.{u1} α] [_inst_3 : GCDMonoid.{u1} α _inst_2] (x : α) (y : α), Exists.{succ u1} α (fun (x' : α) => Exists.{succ u1} α (fun (y' : α) => And (Eq.{succ u1} α x (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) x')) (And (Eq.{succ u1} α y (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) y')) (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x' y')))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_2 : CancelCommMonoidWithZero.{u1} α] [_inst_3 : GCDMonoid.{u1} α _inst_2] (x : α) (y : α), Exists.{succ u1} α (fun (x' : α) => Exists.{succ u1} α (fun (y' : α) => And (Eq.{succ u1} α x (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) x')) (And (Eq.{succ u1} α y (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2)))))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x y) y')) (IsUnit.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_2))) (GCDMonoid.gcd.{u1} α _inst_2 _inst_3 x' y')))))
Case conversion may be inaccurate. Consider using '#align extract_gcd extract_gcdₓ'. -/
theorem extract_gcd {α : Type _} [CancelCommMonoidWithZero α] [GCDMonoid α] (x y : α) :
    ∃ x' y', x = gcd x y * x' ∧ y = gcd x y * y' ∧ IsUnit (gcd x' y') :=
  by
  by_cases h : gcd x y = 0
  · obtain ⟨rfl, rfl⟩ := (gcd_eq_zero_iff x y).1 h
    simp_rw [← associated_one_iff_isUnit]
    exact
      ⟨1, 1, by rw [h, MulZeroClass.zero_mul], by rw [h, MulZeroClass.zero_mul], gcd_one_left' 1⟩
  obtain ⟨x', ex⟩ := gcd_dvd_left x y
  obtain ⟨y', ey⟩ := gcd_dvd_right x y
  exact ⟨x', y', ex, ey, isUnit_gcd_of_eq_mul_gcd ex ey h⟩
#align extract_gcd extract_gcd

end Gcd

section Lcm

#print lcm_dvd_iff /-
theorem lcm_dvd_iff [GCDMonoid α] {a b c : α} : lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c :=
  by
  by_cases this : a = 0 ∨ b = 0
  ·
    rcases this with (rfl | rfl) <;>
      simp (config := { contextual := true }) only [iff_def, lcm_zero_left, lcm_zero_right,
        zero_dvd_iff, dvd_zero, eq_self_iff_true, and_true_iff, imp_true_iff]
  · obtain ⟨h1, h2⟩ := not_or.1 this
    have h : gcd a b ≠ 0 := fun H => h1 ((gcd_eq_zero_iff _ _).1 H).1
    rw [← mul_dvd_mul_iff_left h, (gcd_mul_lcm a b).dvd_iff_dvd_left, ←
      (gcd_mul_right' c a b).dvd_iff_dvd_right, dvd_gcd_iff, mul_comm b c, mul_dvd_mul_iff_left h1,
      mul_dvd_mul_iff_right h2, and_comm']
#align lcm_dvd_iff lcm_dvd_iff
-/

#print dvd_lcm_left /-
theorem dvd_lcm_left [GCDMonoid α] (a b : α) : a ∣ lcm a b :=
  (lcm_dvd_iff.1 (dvd_refl (lcm a b))).1
#align dvd_lcm_left dvd_lcm_left
-/

#print dvd_lcm_right /-
theorem dvd_lcm_right [GCDMonoid α] (a b : α) : b ∣ lcm a b :=
  (lcm_dvd_iff.1 (dvd_refl (lcm a b))).2
#align dvd_lcm_right dvd_lcm_right
-/

#print lcm_dvd /-
theorem lcm_dvd [GCDMonoid α] {a b c : α} (hab : a ∣ b) (hcb : c ∣ b) : lcm a c ∣ b :=
  lcm_dvd_iff.2 ⟨hab, hcb⟩
#align lcm_dvd lcm_dvd
-/

/- warning: lcm_eq_zero_iff -> lcm_eq_zero_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α), Iff (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 a b) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (Or (Eq.{succ u1} α a (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (Eq.{succ u1} α b (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (a : α) (b : α), Iff (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 a b) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (Or (Eq.{succ u1} α a (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (Eq.{succ u1} α b (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))
Case conversion may be inaccurate. Consider using '#align lcm_eq_zero_iff lcm_eq_zero_iffₓ'. -/
@[simp]
theorem lcm_eq_zero_iff [GCDMonoid α] (a b : α) : lcm a b = 0 ↔ a = 0 ∨ b = 0 :=
  Iff.intro
    (fun h : lcm a b = 0 =>
      by
      have : Associated (a * b) 0 :=
        (gcd_mul_lcm a b).symm.trans <| by rw [h, MulZeroClass.mul_zero]
      simpa only [associated_zero_iff_eq_zero, mul_eq_zero] )
    (by rintro (rfl | rfl) <;> [apply lcm_zero_left, apply lcm_zero_right])
#align lcm_eq_zero_iff lcm_eq_zero_iff

/- warning: normalize_lcm -> normalize_lcm is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b)) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b)) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b)
Case conversion may be inaccurate. Consider using '#align normalize_lcm normalize_lcmₓ'. -/
@[simp]
theorem normalize_lcm [NormalizedGCDMonoid α] (a b : α) : normalize (lcm a b) = lcm a b :=
  NormalizedGCDMonoid.normalize_lcm a b
#align normalize_lcm normalize_lcm

#print lcm_comm /-
theorem lcm_comm [NormalizedGCDMonoid α] (a b : α) : lcm a b = lcm b a :=
  dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _)
    (lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _))
    (lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _))
#align lcm_comm lcm_comm
-/

#print lcm_comm' /-
theorem lcm_comm' [GCDMonoid α] (a b : α) : Associated (lcm a b) (lcm b a) :=
  associated_of_dvd_dvd (lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _))
    (lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _))
#align lcm_comm' lcm_comm'
-/

#print lcm_assoc /-
theorem lcm_assoc [NormalizedGCDMonoid α] (m n k : α) : lcm (lcm m n) k = lcm m (lcm n k) :=
  dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _)
    (lcm_dvd (lcm_dvd (dvd_lcm_left _ _) ((dvd_lcm_left _ _).trans (dvd_lcm_right _ _)))
      ((dvd_lcm_right _ _).trans (dvd_lcm_right _ _)))
    (lcm_dvd ((dvd_lcm_left _ _).trans (dvd_lcm_left _ _))
      (lcm_dvd ((dvd_lcm_right _ _).trans (dvd_lcm_left _ _)) (dvd_lcm_right _ _)))
#align lcm_assoc lcm_assoc
-/

#print lcm_assoc' /-
theorem lcm_assoc' [GCDMonoid α] (m n k : α) : Associated (lcm (lcm m n) k) (lcm m (lcm n k)) :=
  associated_of_dvd_dvd
    (lcm_dvd (lcm_dvd (dvd_lcm_left _ _) ((dvd_lcm_left _ _).trans (dvd_lcm_right _ _)))
      ((dvd_lcm_right _ _).trans (dvd_lcm_right _ _)))
    (lcm_dvd ((dvd_lcm_left _ _).trans (dvd_lcm_left _ _))
      (lcm_dvd ((dvd_lcm_right _ _).trans (dvd_lcm_left _ _)) (dvd_lcm_right _ _)))
#align lcm_assoc' lcm_assoc'
-/

instance [NormalizedGCDMonoid α] : IsCommutative α lcm :=
  ⟨lcm_comm⟩

instance [NormalizedGCDMonoid α] : IsAssociative α lcm :=
  ⟨lcm_assoc⟩

/- warning: lcm_eq_normalize -> lcm_eq_normalize is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] {a : α} {b : α} {c : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) c) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b)) -> (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) c))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] {a : α} {b : α} {c : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) c) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b)) -> (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) c))
Case conversion may be inaccurate. Consider using '#align lcm_eq_normalize lcm_eq_normalizeₓ'. -/
theorem lcm_eq_normalize [NormalizedGCDMonoid α] {a b c : α} (habc : lcm a b ∣ c)
    (hcab : c ∣ lcm a b) : lcm a b = normalize c :=
  normalize_lcm a b ▸ normalize_eq_normalize habc hcab
#align lcm_eq_normalize lcm_eq_normalize

#print lcm_dvd_lcm /-
theorem lcm_dvd_lcm [GCDMonoid α] {a b c d : α} (hab : a ∣ b) (hcd : c ∣ d) : lcm a c ∣ lcm b d :=
  lcm_dvd (hab.trans (dvd_lcm_left _ _)) (hcd.trans (dvd_lcm_right _ _))
#align lcm_dvd_lcm lcm_dvd_lcm
-/

/- warning: lcm_units_coe_left -> lcm_units_coe_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (u : Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (a : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (Units.hasCoe.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) u) a) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (u : Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (a : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
Case conversion may be inaccurate. Consider using '#align lcm_units_coe_left lcm_units_coe_leftₓ'. -/
@[simp]
theorem lcm_units_coe_left [NormalizedGCDMonoid α] (u : αˣ) (a : α) : lcm (↑u) a = normalize a :=
  lcm_eq_normalize (lcm_dvd Units.coe_dvd dvd_rfl) (dvd_lcm_right _ _)
#align lcm_units_coe_left lcm_units_coe_left

/- warning: lcm_units_coe_right -> lcm_units_coe_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (u : Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (Units.hasCoe.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) u)) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (u : Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a (Units.val.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) u)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
Case conversion may be inaccurate. Consider using '#align lcm_units_coe_right lcm_units_coe_rightₓ'. -/
@[simp]
theorem lcm_units_coe_right [NormalizedGCDMonoid α] (a : α) (u : αˣ) : lcm a ↑u = normalize a :=
  (lcm_comm a u).trans <| lcm_units_coe_left _ _
#align lcm_units_coe_right lcm_units_coe_right

/- warning: lcm_one_left -> lcm_one_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))))) a) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
Case conversion may be inaccurate. Consider using '#align lcm_one_left lcm_one_leftₓ'. -/
@[simp]
theorem lcm_one_left [NormalizedGCDMonoid α] (a : α) : lcm 1 a = normalize a :=
  lcm_units_coe_left 1 a
#align lcm_one_left lcm_one_left

/- warning: lcm_one_right -> lcm_one_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
Case conversion may be inaccurate. Consider using '#align lcm_one_right lcm_one_rightₓ'. -/
@[simp]
theorem lcm_one_right [NormalizedGCDMonoid α] (a : α) : lcm a 1 = normalize a :=
  lcm_units_coe_right a 1
#align lcm_one_right lcm_one_right

/- warning: lcm_same -> lcm_same is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a a) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a)
Case conversion may be inaccurate. Consider using '#align lcm_same lcm_sameₓ'. -/
@[simp]
theorem lcm_same [NormalizedGCDMonoid α] (a : α) : lcm a a = normalize a :=
  lcm_eq_normalize (lcm_dvd dvd_rfl dvd_rfl) (dvd_lcm_left _ _)
#align lcm_same lcm_same

/- warning: lcm_eq_one_iff -> lcm_eq_one_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), Iff (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (And (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), Iff (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (And (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Monoid.toOne.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
Case conversion may be inaccurate. Consider using '#align lcm_eq_one_iff lcm_eq_one_iffₓ'. -/
@[simp]
theorem lcm_eq_one_iff [NormalizedGCDMonoid α] (a b : α) : lcm a b = 1 ↔ a ∣ 1 ∧ b ∣ 1 :=
  Iff.intro (fun eq => Eq ▸ ⟨dvd_lcm_left _ _, dvd_lcm_right _ _⟩) fun ⟨⟨c, hc⟩, ⟨d, hd⟩⟩ =>
    show lcm (Units.mkOfMulEqOne a c hc.symm : α) (Units.mkOfMulEqOne b d hd.symm) = 1 by
      rw [lcm_units_coe_left, normalize_coe_units]
#align lcm_eq_one_iff lcm_eq_one_iff

/- warning: lcm_mul_left -> lcm_mul_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a c)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) b c))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) a c)) (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) α ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (MulZeroClass.toMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (MulZeroOneClass.toMulZeroClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (MonoidWithZero.toMulZeroOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (CommMonoidWithZero.toMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) _inst_1)))))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) b c))
Case conversion may be inaccurate. Consider using '#align lcm_mul_left lcm_mul_leftₓ'. -/
@[simp]
theorem lcm_mul_left [NormalizedGCDMonoid α] (a b c : α) :
    lcm (a * b) (a * c) = normalize a * lcm b c :=
  by_cases (by rintro rfl <;> simp only [MulZeroClass.zero_mul, lcm_zero_left, normalize_zero])
    fun ha : a ≠ 0 =>
    suffices lcm (a * b) (a * c) = normalize (a * lcm b c) by
      simpa only [normalize.map_mul, normalize_lcm]
    have : a ∣ lcm (a * b) (a * c) := (dvd_mul_right _ _).trans (dvd_lcm_left _ _)
    let ⟨d, Eq⟩ := this
    lcm_eq_normalize
      (lcm_dvd (mul_dvd_mul_left a (dvd_lcm_left _ _)) (mul_dvd_mul_left a (dvd_lcm_right _ _)))
      (Eq.symm ▸
        (mul_dvd_mul_left a <|
          lcm_dvd ((mul_dvd_mul_iff_left ha).1 <| Eq ▸ dvd_lcm_left _ _)
            ((mul_dvd_mul_iff_left ha).1 <| Eq ▸ dvd_lcm_right _ _)))
#align lcm_mul_left lcm_mul_left

/- warning: lcm_mul_right -> lcm_mul_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) b a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) c a)) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) b c) (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α) (c : α), Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) b a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) c a)) (HMul.hMul.{u1, u1, u1} α ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) b c) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a))
Case conversion may be inaccurate. Consider using '#align lcm_mul_right lcm_mul_rightₓ'. -/
@[simp]
theorem lcm_mul_right [NormalizedGCDMonoid α] (a b c : α) :
    lcm (b * a) (c * a) = lcm b c * normalize a := by simp only [mul_comm, lcm_mul_left]
#align lcm_mul_right lcm_mul_right

/- warning: lcm_eq_left_iff -> lcm_eq_left_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a) a) -> (Iff (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) a) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) a) a) -> (Iff (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) a) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a))
Case conversion may be inaccurate. Consider using '#align lcm_eq_left_iff lcm_eq_left_iffₓ'. -/
theorem lcm_eq_left_iff [NormalizedGCDMonoid α] (a b : α) (h : normalize a = a) :
    lcm a b = a ↔ b ∣ a :=
  Iff.intro (fun eq => Eq ▸ dvd_lcm_right _ _) fun hab =>
    dvd_antisymm_of_normalize_eq (normalize_lcm _ _) h (lcm_dvd (dvd_refl a) hab) (dvd_lcm_left _ _)
#align lcm_eq_left_iff lcm_eq_left_iff

/- warning: lcm_eq_right_iff -> lcm_eq_right_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), (Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) b) b) -> (Iff (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGcdMonoid.{u1} α _inst_1 _inst_2) a b) b) (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizedGCDMonoid.{u1} α _inst_1] (a : α) (b : α), (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) b) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (NormalizedGCDMonoid.toNormalizationMonoid.{u1} α _inst_1 _inst_2)) b) b) -> (Iff (Eq.{succ u1} α (GCDMonoid.lcm.{u1} α _inst_1 (NormalizedGCDMonoid.toGCDMonoid.{u1} α _inst_1 _inst_2) a b) b) (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b))
Case conversion may be inaccurate. Consider using '#align lcm_eq_right_iff lcm_eq_right_iffₓ'. -/
theorem lcm_eq_right_iff [NormalizedGCDMonoid α] (a b : α) (h : normalize b = b) :
    lcm a b = b ↔ a ∣ b := by simpa only [lcm_comm b a] using lcm_eq_left_iff b a h
#align lcm_eq_right_iff lcm_eq_right_iff

/- warning: lcm_dvd_lcm_mul_left -> lcm_dvd_lcm_mul_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) k m) n)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) k m) n)
Case conversion may be inaccurate. Consider using '#align lcm_dvd_lcm_mul_left lcm_dvd_lcm_mul_leftₓ'. -/
theorem lcm_dvd_lcm_mul_left [GCDMonoid α] (m n k : α) : lcm m n ∣ lcm (k * m) n :=
  lcm_dvd_lcm (dvd_mul_left _ _) dvd_rfl
#align lcm_dvd_lcm_mul_left lcm_dvd_lcm_mul_left

/- warning: lcm_dvd_lcm_mul_right -> lcm_dvd_lcm_mul_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m k) n)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) m k) n)
Case conversion may be inaccurate. Consider using '#align lcm_dvd_lcm_mul_right lcm_dvd_lcm_mul_rightₓ'. -/
theorem lcm_dvd_lcm_mul_right [GCDMonoid α] (m n k : α) : lcm m n ∣ lcm (m * k) n :=
  lcm_dvd_lcm (dvd_mul_right _ _) dvd_rfl
#align lcm_dvd_lcm_mul_right lcm_dvd_lcm_mul_right

/- warning: lcm_dvd_lcm_mul_left_right -> lcm_dvd_lcm_mul_left_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) k n))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) k n))
Case conversion may be inaccurate. Consider using '#align lcm_dvd_lcm_mul_left_right lcm_dvd_lcm_mul_left_rightₓ'. -/
theorem lcm_dvd_lcm_mul_left_right [GCDMonoid α] (m n k : α) : lcm m n ∣ lcm m (k * n) :=
  lcm_dvd_lcm dvd_rfl (dvd_mul_left _ _)
#align lcm_dvd_lcm_mul_left_right lcm_dvd_lcm_mul_left_right

/- warning: lcm_dvd_lcm_mul_right_right -> lcm_dvd_lcm_mul_right_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) n k))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : GCDMonoid.{u1} α _inst_1] (m : α) (n : α) (k : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m n) (GCDMonoid.lcm.{u1} α _inst_1 _inst_2 m (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) n k))
Case conversion may be inaccurate. Consider using '#align lcm_dvd_lcm_mul_right_right lcm_dvd_lcm_mul_right_rightₓ'. -/
theorem lcm_dvd_lcm_mul_right_right [GCDMonoid α] (m n k : α) : lcm m n ∣ lcm m (n * k) :=
  lcm_dvd_lcm dvd_rfl (dvd_mul_right _ _)
#align lcm_dvd_lcm_mul_right_right lcm_dvd_lcm_mul_right_right

#print lcm_eq_of_associated_left /-
theorem lcm_eq_of_associated_left [NormalizedGCDMonoid α] {m n : α} (h : Associated m n) (k : α) :
    lcm m k = lcm n k :=
  dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _) (lcm_dvd_lcm h.Dvd dvd_rfl)
    (lcm_dvd_lcm h.symm.Dvd dvd_rfl)
#align lcm_eq_of_associated_left lcm_eq_of_associated_left
-/

#print lcm_eq_of_associated_right /-
theorem lcm_eq_of_associated_right [NormalizedGCDMonoid α] {m n : α} (h : Associated m n) (k : α) :
    lcm k m = lcm k n :=
  dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _) (lcm_dvd_lcm dvd_rfl h.Dvd)
    (lcm_dvd_lcm dvd_rfl h.symm.Dvd)
#align lcm_eq_of_associated_right lcm_eq_of_associated_right
-/

end Lcm

namespace GCDMonoid

#print GCDMonoid.prime_of_irreducible /-
theorem prime_of_irreducible [GCDMonoid α] {x : α} (hi : Irreducible x) : Prime x :=
  ⟨hi.NeZero,
    ⟨hi.1, fun a b h => by
      cases' gcd_dvd_left x a with y hy
      cases' hi.is_unit_or_is_unit hy with hu hu
      · right
        trans gcd (x * b) (a * b)
        apply dvd_gcd (dvd_mul_right x b) h
        rw [(gcd_mul_right' b x a).dvd_iff_dvd_left]
        exact (associated_unit_mul_left _ _ hu).Dvd
      · left
        rw [hy]
        exact dvd_trans (associated_mul_unit_left _ _ hu).Dvd (gcd_dvd_right x a)⟩⟩
#align gcd_monoid.prime_of_irreducible GCDMonoid.prime_of_irreducible
-/

#print GCDMonoid.irreducible_iff_prime /-
theorem irreducible_iff_prime [GCDMonoid α] {p : α} : Irreducible p ↔ Prime p :=
  ⟨prime_of_irreducible, Prime.irreducible⟩
#align gcd_monoid.irreducible_iff_prime GCDMonoid.irreducible_iff_prime
-/

end GCDMonoid

end GCDMonoid

section UniqueUnit

variable [CancelCommMonoidWithZero α] [Unique αˣ]

#print normalizationMonoidOfUniqueUnits /-
-- see Note [lower instance priority]
instance (priority := 100) normalizationMonoidOfUniqueUnits : NormalizationMonoid α
    where
  normUnit x := 1
  normUnit_zero := rfl
  normUnit_mul x y hx hy := (mul_one 1).symm
  normUnit_coe_units u := Subsingleton.elim _ _
#align normalization_monoid_of_unique_units normalizationMonoidOfUniqueUnits
-/

#print uniqueNormalizationMonoidOfUniqueUnits /-
instance uniqueNormalizationMonoidOfUniqueUnits : Unique (NormalizationMonoid α)
    where
  default := normalizationMonoidOfUniqueUnits
  uniq := fun ⟨u, _, _, _⟩ => by simpa only [(Subsingleton.elim _ _ : u = fun _ => 1)]
#align unique_normalization_monoid_of_unique_units uniqueNormalizationMonoidOfUniqueUnits
-/

#print subsingleton_gcdMonoid_of_unique_units /-
instance subsingleton_gcdMonoid_of_unique_units : Subsingleton (GCDMonoid α) :=
  ⟨fun g₁ g₂ =>
    by
    have hgcd : g₁.gcd = g₂.gcd := by
      ext (a b)
      refine' associated_iff_eq.mp (associated_of_dvd_dvd _ _) <;>
        apply dvd_gcd (gcd_dvd_left _ _) (gcd_dvd_right _ _)
    have hlcm : g₁.lcm = g₂.lcm := by
      ext (a b)
      refine' associated_iff_eq.mp (associated_of_dvd_dvd _ _) <;>
        apply lcm_dvd_iff.2 ⟨dvd_lcm_left _ _, dvd_lcm_right _ _⟩
    cases g₁
    cases g₂
    dsimp only at hgcd hlcm
    simp only [hgcd, hlcm]⟩
#align subsingleton_gcd_monoid_of_unique_units subsingleton_gcdMonoid_of_unique_units
-/

#print subsingleton_normalizedGCDMonoid_of_unique_units /-
instance subsingleton_normalizedGCDMonoid_of_unique_units : Subsingleton (NormalizedGCDMonoid α) :=
  ⟨by
    intro a b
    cases' a with a_norm a_gcd
    cases' b with b_norm b_gcd
    have := Subsingleton.elim a_gcd b_gcd
    subst this
    have := Subsingleton.elim a_norm b_norm
    subst this⟩
#align subsingleton_normalized_gcd_monoid_of_unique_units subsingleton_normalizedGCDMonoid_of_unique_units
-/

/- warning: norm_unit_eq_one -> normUnit_eq_one is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : Unique.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))] (x : α), Eq.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (NormalizationMonoid.normUnit.{u1} α _inst_1 (normalizationMonoidOfUniqueUnits.{u1} α _inst_1 _inst_2) x) (OfNat.ofNat.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (OfNat.mk.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (One.one.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulOneClass.toHasOne.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Units.mulOneClass.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : Unique.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))] (x : α), Eq.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (NormalizationMonoid.normUnit.{u1} α _inst_1 (normalizationMonoidOfUniqueUnits.{u1} α _inst_1 _inst_2) x) (OfNat.ofNat.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) 1 (One.toOfNat1.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (InvOneClass.toOne.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivInvOneMonoid.toInvOneClass.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivisionMonoid.toDivInvOneMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (DivisionCommMonoid.toDivisionMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (CommGroup.toDivisionCommMonoid.{u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Units.instCommGroupUnitsToMonoid.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))))))
Case conversion may be inaccurate. Consider using '#align norm_unit_eq_one normUnit_eq_oneₓ'. -/
@[simp]
theorem normUnit_eq_one (x : α) : normUnit x = 1 :=
  rfl
#align norm_unit_eq_one normUnit_eq_one

/- warning: normalize_eq -> normalize_eq is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : Unique.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))] (x : α), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 (normalizationMonoidOfUniqueUnits.{u1} α _inst_1 _inst_2)) x) x
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : Unique.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))] (x : α), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) x) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 (normalizationMonoidOfUniqueUnits.{u1} α _inst_1 _inst_2)) x) x
Case conversion may be inaccurate. Consider using '#align normalize_eq normalize_eqₓ'. -/
@[simp]
theorem normalize_eq (x : α) : normalize x = x :=
  mul_one x
#align normalize_eq normalize_eq

/- warning: associates_equiv_of_unique_units -> associatesEquivOfUniqueUnits is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : Unique.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))], MulEquiv.{u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (Associates.hasMul.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MulZeroClass.toHasMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : Unique.{succ u1} (Units.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))], MulEquiv.{u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (Associates.instMulAssociatesToMonoid.{u1} α (CommMonoidWithZero.toCommMonoid.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MulZeroClass.toMul.{u1} α (MulZeroOneClass.toMulZeroClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))
Case conversion may be inaccurate. Consider using '#align associates_equiv_of_unique_units associatesEquivOfUniqueUnitsₓ'. -/
/-- If a monoid's only unit is `1`, then it is isomorphic to its associates. -/
@[simps]
def associatesEquivOfUniqueUnits : Associates α ≃* α
    where
  toFun := Associates.out
  invFun := Associates.mk
  left_inv := Associates.mk_out
  right_inv t := (Associates.out_mk _).trans <| normalize_eq _
  map_mul' := Associates.out_mul
#align associates_equiv_of_unique_units associatesEquivOfUniqueUnits

end UniqueUnit

section IsDomain

variable [CommRing α] [IsDomain α] [NormalizedGCDMonoid α]

/- warning: gcd_eq_of_dvd_sub_right -> gcd_eq_of_dvd_sub_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CommRing.{u1} α] [_inst_2 : IsDomain.{u1} α (Ring.toSemiring.{u1} α (CommRing.toRing.{u1} α _inst_1))] [_inst_3 : NormalizedGCDMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2)] {a : α} {b : α} {c : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (NonUnitalSemiring.toSemigroupWithZero.{u1} α (NonUnitalRing.toNonUnitalSemiring.{u1} α (NonUnitalCommRing.toNonUnitalRing.{u1} α (CommRing.toNonUnitalCommRing.{u1} α _inst_1)))))) a (HSub.hSub.{u1, u1, u1} α α α (instHSub.{u1} α (SubNegMonoid.toHasSub.{u1} α (AddGroup.toSubNegMonoid.{u1} α (AddGroupWithOne.toAddGroup.{u1} α (AddCommGroupWithOne.toAddGroupWithOne.{u1} α (Ring.toAddCommGroupWithOne.{u1} α (CommRing.toRing.{u1} α _inst_1))))))) b c)) -> (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) (NormalizedGCDMonoid.toGcdMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) _inst_3) a b) (GCDMonoid.gcd.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) (NormalizedGCDMonoid.toGcdMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) _inst_3) a c))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CommRing.{u1} α] [_inst_2 : IsDomain.{u1} α (Ring.toSemiring.{u1} α (CommRing.toRing.{u1} α _inst_1))] [_inst_3 : NormalizedGCDMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2)] {a : α} {b : α} {c : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (NonUnitalSemiring.toSemigroupWithZero.{u1} α (NonUnitalRing.toNonUnitalSemiring.{u1} α (NonUnitalCommRing.toNonUnitalRing.{u1} α (CommRing.toNonUnitalCommRing.{u1} α _inst_1)))))) a (HSub.hSub.{u1, u1, u1} α α α (instHSub.{u1} α (Ring.toSub.{u1} α (CommRing.toRing.{u1} α _inst_1))) b c)) -> (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) (NormalizedGCDMonoid.toGCDMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) _inst_3) a b) (GCDMonoid.gcd.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) (NormalizedGCDMonoid.toGCDMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) _inst_3) a c))
Case conversion may be inaccurate. Consider using '#align gcd_eq_of_dvd_sub_right gcd_eq_of_dvd_sub_rightₓ'. -/
theorem gcd_eq_of_dvd_sub_right {a b c : α} (h : a ∣ b - c) : gcd a b = gcd a c :=
  by
  apply dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _) <;>
      rw [dvd_gcd_iff] <;>
    refine' ⟨gcd_dvd_left _ _, _⟩
  · rcases h with ⟨d, hd⟩
    rcases gcd_dvd_right a b with ⟨e, he⟩
    rcases gcd_dvd_left a b with ⟨f, hf⟩
    use e - f * d
    rw [mul_sub, ← he, ← mul_assoc, ← hf, ← hd, sub_sub_cancel]
  · rcases h with ⟨d, hd⟩
    rcases gcd_dvd_right a c with ⟨e, he⟩
    rcases gcd_dvd_left a c with ⟨f, hf⟩
    use e + f * d
    rw [mul_add, ← he, ← mul_assoc, ← hf, ← hd, ← add_sub_assoc, add_comm c b, add_sub_cancel]
#align gcd_eq_of_dvd_sub_right gcd_eq_of_dvd_sub_right

/- warning: gcd_eq_of_dvd_sub_left -> gcd_eq_of_dvd_sub_left is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CommRing.{u1} α] [_inst_2 : IsDomain.{u1} α (Ring.toSemiring.{u1} α (CommRing.toRing.{u1} α _inst_1))] [_inst_3 : NormalizedGCDMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2)] {a : α} {b : α} {c : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (NonUnitalSemiring.toSemigroupWithZero.{u1} α (NonUnitalRing.toNonUnitalSemiring.{u1} α (NonUnitalCommRing.toNonUnitalRing.{u1} α (CommRing.toNonUnitalCommRing.{u1} α _inst_1)))))) a (HSub.hSub.{u1, u1, u1} α α α (instHSub.{u1} α (SubNegMonoid.toHasSub.{u1} α (AddGroup.toSubNegMonoid.{u1} α (AddGroupWithOne.toAddGroup.{u1} α (AddCommGroupWithOne.toAddGroupWithOne.{u1} α (Ring.toAddCommGroupWithOne.{u1} α (CommRing.toRing.{u1} α _inst_1))))))) b c)) -> (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) (NormalizedGCDMonoid.toGcdMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) _inst_3) b a) (GCDMonoid.gcd.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) (NormalizedGCDMonoid.toGcdMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) _inst_3) c a))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CommRing.{u1} α] [_inst_2 : IsDomain.{u1} α (Ring.toSemiring.{u1} α (CommRing.toRing.{u1} α _inst_1))] [_inst_3 : NormalizedGCDMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2)] {a : α} {b : α} {c : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (NonUnitalSemiring.toSemigroupWithZero.{u1} α (NonUnitalRing.toNonUnitalSemiring.{u1} α (NonUnitalCommRing.toNonUnitalRing.{u1} α (CommRing.toNonUnitalCommRing.{u1} α _inst_1)))))) a (HSub.hSub.{u1, u1, u1} α α α (instHSub.{u1} α (Ring.toSub.{u1} α (CommRing.toRing.{u1} α _inst_1))) b c)) -> (Eq.{succ u1} α (GCDMonoid.gcd.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) (NormalizedGCDMonoid.toGCDMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) _inst_3) b a) (GCDMonoid.gcd.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) (NormalizedGCDMonoid.toGCDMonoid.{u1} α (IsDomain.toCancelCommMonoidWithZero.{u1} α (CommRing.toCommSemiring.{u1} α _inst_1) _inst_2) _inst_3) c a))
Case conversion may be inaccurate. Consider using '#align gcd_eq_of_dvd_sub_left gcd_eq_of_dvd_sub_leftₓ'. -/
theorem gcd_eq_of_dvd_sub_left {a b c : α} (h : a ∣ b - c) : gcd b a = gcd c a := by
  rw [gcd_comm _ a, gcd_comm _ a, gcd_eq_of_dvd_sub_right h]
#align gcd_eq_of_dvd_sub_left gcd_eq_of_dvd_sub_left

end IsDomain

section Constructors

noncomputable section

open Associates

variable [CancelCommMonoidWithZero α]

private theorem map_mk_unit_aux [DecidableEq α] {f : Associates α →* α}
    (hinv : Function.RightInverse f Associates.mk) (a : α) :
    a * ↑(Classical.choose (associated_map_mk hinv a)) = f (Associates.mk a) :=
  Classical.choose_spec (associated_map_mk hinv a)
#align map_mk_unit_aux map_mk_unit_aux

/- warning: normalization_monoid_of_monoid_hom_right_inverse -> normalizationMonoidOfMonoidHomRightInverse is a dubious translation:
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(MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))), (Function.RightInverse.{succ u1, succ u1} α (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (coeFn.{succ u1, succ u1} (MonoidHom.{u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (MulZeroOneClass.toMulOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.commMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (fun (_x : MonoidHom.{u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (MulZeroOneClass.toMulOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α 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(CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) -> (NormalizationMonoid.{u1} α _inst_1)
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(CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))), (Function.RightInverse.{succ u1, succ u1} α (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (MulZeroOneClass.toMulOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α 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(CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (MulOneClass.toMul.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (MulZeroOneClass.toMulOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (MulZeroOneClass.toMulOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidHom.monoidHomClass.{u1, u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (MulZeroOneClass.toMulOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) f) (Associates.mk.{u1} α (MonoidWithZero.toMonoid.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) -> (NormalizationMonoid.{u1} α _inst_1)
Case conversion may be inaccurate. Consider using '#align normalization_monoid_of_monoid_hom_right_inverse normalizationMonoidOfMonoidHomRightInverseₓ'. -/
/-- Define `normalization_monoid` on a structure from a `monoid_hom` inverse to `associates.mk`. -/
def normalizationMonoidOfMonoidHomRightInverse [DecidableEq α] (f : Associates α →* α)
    (hinv : Function.RightInverse f Associates.mk) : NormalizationMonoid α
    where
  normUnit a :=
    if a = 0 then 1
    else Classical.choose (Associates.mk_eq_mk_iff_associated.1 (hinv (Associates.mk a)).symm)
  normUnit_zero := if_pos rfl
  normUnit_mul a b ha hb :=
    by
    rw [if_neg (mul_ne_zero ha hb), if_neg ha, if_neg hb, Units.ext_iff, Units.val_mul]
    suffices
      a * b * ↑(Classical.choose (associated_map_mk hinv (a * b))) =
        a * ↑(Classical.choose (associated_map_mk hinv a)) *
          (b * ↑(Classical.choose (associated_map_mk hinv b)))
      by
      apply mul_left_cancel₀ (mul_ne_zero ha hb) _
      simpa only [mul_assoc, mul_comm, mul_left_comm] using this
    rw [map_mk_unit_aux hinv a, map_mk_unit_aux hinv (a * b), map_mk_unit_aux hinv b, ←
      MonoidHom.map_mul, Associates.mk_mul_mk]
  normUnit_coe_units u := by
    nontriviality α
    rw [if_neg (Units.ne_zero u), Units.ext_iff]
    apply mul_left_cancel₀ (Units.ne_zero u)
    rw [Units.mul_inv, map_mk_unit_aux hinv u,
      Associates.mk_eq_mk_iff_associated.2 (associated_one_iff_isUnit.2 ⟨u, rfl⟩),
      Associates.mk_one, MonoidHom.map_one]
#align normalization_monoid_of_monoid_hom_right_inverse normalizationMonoidOfMonoidHomRightInverse

#print gcdMonoidOfGCD /-
/-- Define `gcd_monoid` on a structure just from the `gcd` and its properties. -/
noncomputable def gcdMonoidOfGCD [DecidableEq α] (gcd : α → α → α)
    (gcd_dvd_left : ∀ a b, gcd a b ∣ a) (gcd_dvd_right : ∀ a b, gcd a b ∣ b)
    (dvd_gcd : ∀ {a b c}, a ∣ c → a ∣ b → a ∣ gcd c b) : GCDMonoid α :=
  { gcd
    gcd_dvd_left
    gcd_dvd_right
    dvd_gcd := fun a b c => dvd_gcd
    lcm := fun a b =>
      if a = 0 then 0 else Classical.choose ((gcd_dvd_left a b).trans (Dvd.intro b rfl))
    gcd_mul_lcm := fun a b => by
      split_ifs with a0
      · rw [MulZeroClass.mul_zero, a0, MulZeroClass.zero_mul]
      · rw [← Classical.choose_spec ((gcd_dvd_left a b).trans (Dvd.intro b rfl))]
    lcm_zero_left := fun a => if_pos rfl
    lcm_zero_right := fun a => by
      split_ifs with a0
      · rfl
      have h := (Classical.choose_spec ((gcd_dvd_left a 0).trans (Dvd.intro 0 rfl))).symm
      have a0' : gcd a 0 ≠ 0 := by
        contrapose! a0
        rw [← associated_zero_iff_eq_zero, ← a0]
        exact associated_of_dvd_dvd (dvd_gcd (dvd_refl a) (dvd_zero a)) (gcd_dvd_left _ _)
      apply Or.resolve_left (mul_eq_zero.1 _) a0'
      rw [h, MulZeroClass.mul_zero] }
#align gcd_monoid_of_gcd gcdMonoidOfGCD
-/

/- warning: normalized_gcd_monoid_of_gcd -> normalizedGCDMonoidOfGCD is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] [_inst_3 : DecidableEq.{succ u1} α] (gcd : α -> α -> α), (forall (a : α) (b : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (gcd a b) a) -> (forall (a : α) (b : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (gcd a b) b) -> (forall {a : α} {b : α} {c : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a c) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (gcd c b))) -> (forall (a : α) (b : α), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) (gcd a b)) (gcd a b)) -> (NormalizedGCDMonoid.{u1} α _inst_1)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] [_inst_3 : DecidableEq.{succ u1} α] (gcd : α -> α -> α), (forall (a : α) (b : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (gcd a b) a) -> (forall (a : α) (b : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (gcd a b) b) -> (forall {a : α} {b : α} {c : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a c) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a b) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (gcd c b))) -> (forall (a : α) (b : α), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (gcd a b)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) (gcd a b)) (gcd a b)) -> (NormalizedGCDMonoid.{u1} α _inst_1)
Case conversion may be inaccurate. Consider using '#align normalized_gcd_monoid_of_gcd normalizedGCDMonoidOfGCDₓ'. -/
/-- Define `normalized_gcd_monoid` on a structure just from the `gcd` and its properties. -/
noncomputable def normalizedGCDMonoidOfGCD [NormalizationMonoid α] [DecidableEq α] (gcd : α → α → α)
    (gcd_dvd_left : ∀ a b, gcd a b ∣ a) (gcd_dvd_right : ∀ a b, gcd a b ∣ b)
    (dvd_gcd : ∀ {a b c}, a ∣ c → a ∣ b → a ∣ gcd c b)
    (normalize_gcd : ∀ a b, normalize (gcd a b) = gcd a b) : NormalizedGCDMonoid α :=
  { (inferInstance : NormalizationMonoid α) with
    gcd
    gcd_dvd_left
    gcd_dvd_right
    dvd_gcd := fun a b c => dvd_gcd
    normalize_gcd
    lcm := fun a b =>
      if a = 0 then 0
      else Classical.choose (dvd_normalize_iff.2 ((gcd_dvd_left a b).trans (Dvd.intro b rfl)))
    normalize_lcm := fun a b => by
      dsimp [normalize]
      split_ifs with a0
      · exact @normalize_zero α _ _
      · have :=
          (Classical.choose_spec
              (dvd_normalize_iff.2 ((gcd_dvd_left a b).trans (Dvd.intro b rfl)))).symm
        set l := Classical.choose (dvd_normalize_iff.2 ((gcd_dvd_left a b).trans (Dvd.intro b rfl)))
        obtain rfl | hb := eq_or_ne b 0
        · simp only [normalize_zero, MulZeroClass.mul_zero, mul_eq_zero] at this
          obtain ha | hl := this
          · apply (a0 _).elim
            rw [← zero_dvd_iff, ← ha]
            exact gcd_dvd_left _ _
          · convert@normalize_zero α _ _
        have h1 : gcd a b ≠ 0 := by
          have hab : a * b ≠ 0 := mul_ne_zero a0 hb
          contrapose! hab
          rw [← normalize_eq_zero, ← this, hab, MulZeroClass.zero_mul]
        have h2 : normalize (gcd a b * l) = gcd a b * l := by rw [this, normalize_idem]
        rw [← normalize_gcd] at this
        rwa [normalize.map_mul, normalize_gcd, mul_right_inj' h1] at h2
    gcd_mul_lcm := fun a b => by
      split_ifs with a0
      · rw [MulZeroClass.mul_zero, a0, MulZeroClass.zero_mul]
      · rw [←
          Classical.choose_spec (dvd_normalize_iff.2 ((gcd_dvd_left a b).trans (Dvd.intro b rfl)))]
        exact normalize_associated (a * b)
    lcm_zero_left := fun a => if_pos rfl
    lcm_zero_right := fun a => by
      split_ifs with a0
      · rfl
      rw [← normalize_eq_zero] at a0
      have h :=
        (Classical.choose_spec
            (dvd_normalize_iff.2 ((gcd_dvd_left a 0).trans (Dvd.intro 0 rfl)))).symm
      have gcd0 : gcd a 0 = normalize a :=
        by
        rw [← normalize_gcd]
        exact normalize_eq_normalize (gcd_dvd_left _ _) (dvd_gcd (dvd_refl a) (dvd_zero a))
      rw [← gcd0] at a0
      apply Or.resolve_left (mul_eq_zero.1 _) a0
      rw [h, MulZeroClass.mul_zero, normalize_zero] }
#align normalized_gcd_monoid_of_gcd normalizedGCDMonoidOfGCD

#print gcdMonoidOfLCM /-
/-- Define `gcd_monoid` on a structure just from the `lcm` and its properties. -/
noncomputable def gcdMonoidOfLCM [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a) : GCDMonoid α :=
  let exists_gcd a b := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
  { lcm
    gcd := fun a b => if a = 0 then b else if b = 0 then a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      split_ifs
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), MulZeroClass.mul_zero, MulZeroClass.zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), MulZeroClass.mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
    lcm_zero_left := fun a => eq_zero_of_zero_dvd (dvd_lcm_left _ _)
    lcm_zero_right := fun a => eq_zero_of_zero_dvd (dvd_lcm_right _ _)
    gcd_dvd_left := fun a b => by
      split_ifs with h h_1
      · rw [h]
        apply dvd_zero
      · exact dvd_rfl
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [Con, zero_dvd_iff, mul_eq_zero] at h
        cases h <;> tauto
      rw [← mul_dvd_mul_iff_left h0, ← Classical.choose_spec (exists_gcd a b), mul_comm,
        mul_dvd_mul_iff_right h]
      apply dvd_lcm_right
    gcd_dvd_right := fun a b => by
      split_ifs with h h_1
      · exact dvd_rfl
      · rw [h_1]
        apply dvd_zero
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [Con, zero_dvd_iff, mul_eq_zero] at h
        cases h <;> tauto
      rw [← mul_dvd_mul_iff_left h0, ← Classical.choose_spec (exists_gcd a b),
        mul_dvd_mul_iff_right h_1]
      apply dvd_lcm_left
    dvd_gcd := fun a b c ac ab => by
      split_ifs
      · exact ab
      · exact ac
      have h0 : lcm c b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left c rfl)
        rw [Con, zero_dvd_iff, mul_eq_zero] at h
        cases h <;> tauto
      rw [← mul_dvd_mul_iff_left h0, ← Classical.choose_spec (exists_gcd c b)]
      rcases ab with ⟨d, rfl⟩
      rw [mul_eq_zero] at h_1
      push_neg  at h_1
      rw [mul_comm a, ← mul_assoc, mul_dvd_mul_iff_right h_1.1]
      apply lcm_dvd (Dvd.intro d rfl)
      rw [mul_comm, mul_dvd_mul_iff_right h_1.2]
      apply ac }
#align gcd_monoid_of_lcm gcdMonoidOfLCM
-/

/- warning: normalized_gcd_monoid_of_lcm -> normalizedGCDMonoidOfLCM is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] [_inst_3 : DecidableEq.{succ u1} α] (lcm : α -> α -> α), (forall (a : α) (b : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (lcm a b)) -> (forall (a : α) (b : α), Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b (lcm a b)) -> (forall {a : α} {b : α} {c : α}, (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c a) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a) -> (Dvd.Dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (lcm c b) a)) -> (forall (a : α) (b : α), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (fun (_x : MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) => α -> α) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (normalize.{u1} α _inst_1 _inst_2) (lcm a b)) (lcm a b)) -> (NormalizedGCDMonoid.{u1} α _inst_1)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} α] [_inst_2 : NormalizationMonoid.{u1} α _inst_1] [_inst_3 : DecidableEq.{succ u1} α] (lcm : α -> α -> α), (forall (a : α) (b : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) a (lcm a b)) -> (forall (a : α) (b : α), Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b (lcm a b)) -> (forall {a : α} {b : α} {c : α}, (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) c a) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) b a) -> (Dvd.dvd.{u1} α (semigroupDvd.{u1} α (SemigroupWithZero.toSemigroup.{u1} α (MonoidWithZero.toSemigroupWithZero.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (lcm c b) a)) -> (forall (a : α) (b : α), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) (lcm a b)) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => α) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MulZeroOneClass.toMulOneClass.{u1} α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1)))) α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} α α (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))) (MonoidWithZero.toMulZeroOneClass.{u1} α (CommMonoidWithZero.toMonoidWithZero.{u1} α (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} α _inst_1))))))) (normalize.{u1} α _inst_1 _inst_2) (lcm a b)) (lcm a b)) -> (NormalizedGCDMonoid.{u1} α _inst_1)
Case conversion may be inaccurate. Consider using '#align normalized_gcd_monoid_of_lcm normalizedGCDMonoidOfLCMₓ'. -/
/-- Define `normalized_gcd_monoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), MulZeroClass.mul_zero, MulZeroClass.zero_mul]
      ·
        rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), MulZeroClass.mul_zero,
          MulZeroClass.mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [Con, zero_dvd_iff, mul_eq_zero] at h
        cases h <;> tauto
      apply mul_left_cancel₀ h0
      refine' trans _ (Classical.choose_spec (exists_gcd a b))
      conv_lhs =>
        congr
        rw [← normalize_lcm a b]
      erw [← normalize.map_mul, ← Classical.choose_spec (exists_gcd a b), normalize_idem]
    lcm_zero_left := fun a => eq_zero_of_zero_dvd (dvd_lcm_left _ _)
    lcm_zero_right := fun a => eq_zero_of_zero_dvd (dvd_lcm_right _ _)
    gcd_dvd_left := fun a b => by
      split_ifs
      · rw [h]
        apply dvd_zero
      · exact (normalize_associated _).Dvd
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [Con, zero_dvd_iff, mul_eq_zero] at h
        cases h <;> tauto
      rw [← mul_dvd_mul_iff_left h0, ← Classical.choose_spec (exists_gcd a b), normalize_dvd_iff,
        mul_comm, mul_dvd_mul_iff_right h]
      apply dvd_lcm_right
    gcd_dvd_right := fun a b => by
      split_ifs
      · exact (normalize_associated _).Dvd
      · rw [h_1]
        apply dvd_zero
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [Con, zero_dvd_iff, mul_eq_zero] at h
        cases h <;> tauto
      rw [← mul_dvd_mul_iff_left h0, ← Classical.choose_spec (exists_gcd a b), normalize_dvd_iff,
        mul_dvd_mul_iff_right h_1]
      apply dvd_lcm_left
    dvd_gcd := fun a b c ac ab => by
      split_ifs
      · apply dvd_normalize_iff.2 ab
      · apply dvd_normalize_iff.2 ac
      have h0 : lcm c b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left c rfl)
        rw [Con, zero_dvd_iff, mul_eq_zero] at h
        cases h <;> tauto
      rw [← mul_dvd_mul_iff_left h0, ←
        Classical.choose_spec
          (dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left c rfl))),
        dvd_normalize_iff]
      rcases ab with ⟨d, rfl⟩
      rw [mul_eq_zero] at h_1
      push_neg  at h_1
      rw [mul_comm a, ← mul_assoc, mul_dvd_mul_iff_right h_1.1]
      apply lcm_dvd (Dvd.intro d rfl)
      rw [mul_comm, mul_dvd_mul_iff_right h_1.2]
      apply ac }
#align normalized_gcd_monoid_of_lcm normalizedGCDMonoidOfLCM

#print gcdMonoidOfExistsGCD /-
/-- Define a `gcd_monoid` structure on a monoid just from the existence of a `gcd`. -/
noncomputable def gcdMonoidOfExistsGCD [DecidableEq α]
    (h : ∀ a b : α, ∃ c : α, ∀ d : α, d ∣ a ∧ d ∣ b ↔ d ∣ c) : GCDMonoid α :=
  gcdMonoidOfGCD (fun a b => Classical.choose (h a b))
    (fun a b => ((Classical.choose_spec (h a b) (Classical.choose (h a b))).2 dvd_rfl).1)
    (fun a b => ((Classical.choose_spec (h a b) (Classical.choose (h a b))).2 dvd_rfl).2)
    fun a b c ac ab => (Classical.choose_spec (h c b) a).1 ⟨ac, ab⟩
#align gcd_monoid_of_exists_gcd gcdMonoidOfExistsGCD
-/

#print normalizedGCDMonoidOfExistsGCD /-
/-- Define a `normalized_gcd_monoid` structure on a monoid just from the existence of a `gcd`. -/
noncomputable def normalizedGCDMonoidOfExistsGCD [NormalizationMonoid α] [DecidableEq α]
    (h : ∀ a b : α, ∃ c : α, ∀ d : α, d ∣ a ∧ d ∣ b ↔ d ∣ c) : NormalizedGCDMonoid α :=
  normalizedGCDMonoidOfGCD (fun a b => normalize (Classical.choose (h a b)))
    (fun a b =>
      normalize_dvd_iff.2 ((Classical.choose_spec (h a b) (Classical.choose (h a b))).2 dvd_rfl).1)
    (fun a b =>
      normalize_dvd_iff.2 ((Classical.choose_spec (h a b) (Classical.choose (h a b))).2 dvd_rfl).2)
    (fun a b c ac ab => dvd_normalize_iff.2 ((Classical.choose_spec (h c b) a).1 ⟨ac, ab⟩))
    fun a b => normalize_idem _
#align normalized_gcd_monoid_of_exists_gcd normalizedGCDMonoidOfExistsGCD
-/

#print gcdMonoidOfExistsLCM /-
/-- Define a `gcd_monoid` structure on a monoid just from the existence of an `lcm`. -/
noncomputable def gcdMonoidOfExistsLCM [DecidableEq α]
    (h : ∀ a b : α, ∃ c : α, ∀ d : α, a ∣ d ∧ b ∣ d ↔ c ∣ d) : GCDMonoid α :=
  gcdMonoidOfLCM (fun a b => Classical.choose (h a b))
    (fun a b => ((Classical.choose_spec (h a b) (Classical.choose (h a b))).2 dvd_rfl).1)
    (fun a b => ((Classical.choose_spec (h a b) (Classical.choose (h a b))).2 dvd_rfl).2)
    fun a b c ac ab => (Classical.choose_spec (h c b) a).1 ⟨ac, ab⟩
#align gcd_monoid_of_exists_lcm gcdMonoidOfExistsLCM
-/

#print normalizedGCDMonoidOfExistsLCM /-
/-- Define a `normalized_gcd_monoid` structure on a monoid just from the existence of an `lcm`. -/
noncomputable def normalizedGCDMonoidOfExistsLCM [NormalizationMonoid α] [DecidableEq α]
    (h : ∀ a b : α, ∃ c : α, ∀ d : α, a ∣ d ∧ b ∣ d ↔ c ∣ d) : NormalizedGCDMonoid α :=
  normalizedGCDMonoidOfLCM (fun a b => normalize (Classical.choose (h a b)))
    (fun a b =>
      dvd_normalize_iff.2 ((Classical.choose_spec (h a b) (Classical.choose (h a b))).2 dvd_rfl).1)
    (fun a b =>
      dvd_normalize_iff.2 ((Classical.choose_spec (h a b) (Classical.choose (h a b))).2 dvd_rfl).2)
    (fun a b c ac ab => normalize_dvd_iff.2 ((Classical.choose_spec (h c b) a).1 ⟨ac, ab⟩))
    fun a b => normalize_idem _
#align normalized_gcd_monoid_of_exists_lcm normalizedGCDMonoidOfExistsLCM
-/

end Constructors

namespace CommGroupWithZero

variable (G₀ : Type _) [CommGroupWithZero G₀] [DecidableEq G₀]

-- see Note [lower instance priority]
instance (priority := 100) : NormalizedGCDMonoid G₀
    where
  normUnit x := if h : x = 0 then 1 else (Units.mk0 x h)⁻¹
  normUnit_zero := dif_pos rfl
  normUnit_mul x y x0 y0 := Units.eq_iff.1 (by simp [x0, y0, mul_comm])
  normUnit_coe_units u := by
    rw [dif_neg (Units.ne_zero _), Units.mk0_val]
    infer_instance
  gcd a b := if a = 0 ∧ b = 0 then 0 else 1
  lcm a b := if a = 0 ∨ b = 0 then 0 else 1
  gcd_dvd_left a b := by
    split_ifs with h
    · rw [h.1]
    · exact one_dvd _
  gcd_dvd_right a b := by
    split_ifs with h
    · rw [h.2]
    · exact one_dvd _
  dvd_gcd a b c hac hab := by
    split_ifs with h; · apply dvd_zero
    cases' not_and_distrib.mp h with h h <;>
        refine' is_unit_iff_dvd_one.mp (isUnit_of_dvd_unit _ (IsUnit.mk0 _ h)) <;>
      assumption
  gcd_mul_lcm a b := by
    by_cases ha : a = 0; · simp [ha]
    by_cases hb : b = 0; · simp [hb]
    rw [if_neg (not_and_of_not_left _ ha), one_mul, if_neg (not_or_of_not ha hb)]
    exact (associated_one_iff_is_unit.mpr ((IsUnit.mk0 _ ha).mul (IsUnit.mk0 _ hb))).symm
  lcm_zero_left b := if_pos (Or.inl rfl)
  lcm_zero_right a := if_pos (Or.inr rfl)
  -- `split_ifs` wants to split `normalize`, so handle the cases manually
  normalize_gcd a b := if h : a = 0 ∧ b = 0 then by simp [if_pos h] else by simp [if_neg h]
  normalize_lcm a b := if h : a = 0 ∨ b = 0 then by simp [if_pos h] else by simp [if_neg h]

/- warning: comm_group_with_zero.coe_norm_unit -> CommGroupWithZero.coe_normUnit is a dubious translation:
lean 3 declaration is
  forall (G₀ : Type.{u1}) [_inst_1 : CommGroupWithZero.{u1} G₀] [_inst_2 : DecidableEq.{succ u1} G₀] {a : G₀}, (Ne.{succ u1} G₀ a (OfNat.ofNat.{u1} G₀ 0 (OfNat.mk.{u1} G₀ 0 (Zero.zero.{u1} G₀ (MulZeroClass.toHasZero.{u1} G₀ (MulZeroOneClass.toMulZeroClass.{u1} G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (GroupWithZero.toMonoidWithZero.{u1} G₀ (CommGroupWithZero.toGroupWithZero.{u1} G₀ _inst_1))))))))) -> (Eq.{succ u1} G₀ ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} G₀ (MonoidWithZero.toMonoid.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) G₀ (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} G₀ (MonoidWithZero.toMonoid.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) G₀ (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} G₀ (MonoidWithZero.toMonoid.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) G₀ (coeBase.{succ u1, succ u1} (Units.{u1} G₀ (MonoidWithZero.toMonoid.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) G₀ (Units.hasCoe.{u1} G₀ (MonoidWithZero.toMonoid.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))))))) (NormalizationMonoid.normUnit.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1) (NormalizedGCDMonoid.toNormalizationMonoid.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1) (CommGroupWithZero.normalizedGcdMonoid.{u1} G₀ _inst_1 (fun (a : G₀) (b : G₀) => _inst_2 a b))) a)) (Inv.inv.{u1} G₀ (DivInvMonoid.toHasInv.{u1} G₀ (GroupWithZero.toDivInvMonoid.{u1} G₀ (CommGroupWithZero.toGroupWithZero.{u1} G₀ _inst_1))) a))
but is expected to have type
  forall (G₀ : Type.{u1}) [_inst_1 : CommGroupWithZero.{u1} G₀] [_inst_2 : DecidableEq.{succ u1} G₀] {a : G₀}, (Ne.{succ u1} G₀ a (OfNat.ofNat.{u1} G₀ 0 (Zero.toOfNat0.{u1} G₀ (CommMonoidWithZero.toZero.{u1} G₀ (CommGroupWithZero.toCommMonoidWithZero.{u1} G₀ _inst_1))))) -> (Eq.{succ u1} G₀ (Units.val.{u1} G₀ (MonoidWithZero.toMonoid.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (NormalizationMonoid.normUnit.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1) (NormalizedGCDMonoid.toNormalizationMonoid.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1) (CommGroupWithZero.instNormalizedGCDMonoidCancelCommMonoidWithZero.{u1} G₀ _inst_1 (fun (a : G₀) (b : G₀) => _inst_2 a b))) a)) (Inv.inv.{u1} G₀ (CommGroupWithZero.toInv.{u1} G₀ _inst_1) a))
Case conversion may be inaccurate. Consider using '#align comm_group_with_zero.coe_norm_unit CommGroupWithZero.coe_normUnitₓ'. -/
@[simp]
theorem coe_normUnit {a : G₀} (h0 : a ≠ 0) : (↑(normUnit a) : G₀) = a⁻¹ := by simp [norm_unit, h0]
#align comm_group_with_zero.coe_norm_unit CommGroupWithZero.coe_normUnit

/- warning: comm_group_with_zero.normalize_eq_one -> CommGroupWithZero.normalize_eq_one is a dubious translation:
lean 3 declaration is
  forall (G₀ : Type.{u1}) [_inst_1 : CommGroupWithZero.{u1} G₀] [_inst_2 : DecidableEq.{succ u1} G₀] {a : G₀}, (Ne.{succ u1} G₀ a (OfNat.ofNat.{u1} G₀ 0 (OfNat.mk.{u1} G₀ 0 (Zero.zero.{u1} G₀ (MulZeroClass.toHasZero.{u1} G₀ (MulZeroOneClass.toMulZeroClass.{u1} G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (GroupWithZero.toMonoidWithZero.{u1} G₀ (CommGroupWithZero.toGroupWithZero.{u1} G₀ _inst_1))))))))) -> (Eq.{succ u1} G₀ (coeFn.{succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) (fun (_x : MonoidWithZeroHom.{u1, u1} G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) => G₀ -> G₀) (MonoidWithZeroHom.hasCoeToFun.{u1, u1} G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) (normalize.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1) (NormalizedGCDMonoid.toNormalizationMonoid.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1) (CommGroupWithZero.normalizedGcdMonoid.{u1} G₀ _inst_1 (fun (a : G₀) (b : G₀) => _inst_2 a b)))) a) (OfNat.ofNat.{u1} G₀ 1 (OfNat.mk.{u1} G₀ 1 (One.one.{u1} G₀ (MulOneClass.toHasOne.{u1} G₀ (MulZeroOneClass.toMulOneClass.{u1} G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (GroupWithZero.toMonoidWithZero.{u1} G₀ (CommGroupWithZero.toGroupWithZero.{u1} G₀ _inst_1)))))))))
but is expected to have type
  forall (G₀ : Type.{u1}) [_inst_1 : CommGroupWithZero.{u1} G₀] [_inst_2 : DecidableEq.{succ u1} G₀] {a : G₀}, (Ne.{succ u1} G₀ a (OfNat.ofNat.{u1} G₀ 0 (Zero.toOfNat0.{u1} G₀ (CommMonoidWithZero.toZero.{u1} G₀ (CommGroupWithZero.toCommMonoidWithZero.{u1} G₀ _inst_1))))) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) a) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidWithZeroHom.{u1, u1} G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) G₀ (fun (_x : G₀) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) G₀ G₀ (MulOneClass.toMul.{u1} G₀ (MulZeroOneClass.toMulOneClass.{u1} G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))))) (MulOneClass.toMul.{u1} G₀ (MulZeroOneClass.toMulOneClass.{u1} G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) G₀ G₀ (MulZeroOneClass.toMulOneClass.{u1} G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) (MulZeroOneClass.toMulOneClass.{u1} G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u1, u1} (MonoidWithZeroHom.{u1, u1} G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1))))) G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZeroHom.monoidWithZeroHomClass.{u1, u1} G₀ G₀ (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))) (MonoidWithZero.toMulZeroOneClass.{u1} G₀ (CommMonoidWithZero.toMonoidWithZero.{u1} G₀ (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1)))))))) (normalize.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1) (NormalizedGCDMonoid.toNormalizationMonoid.{u1} G₀ (CommGroupWithZero.cancelCommMonoidWithZero.{u1} G₀ _inst_1) (CommGroupWithZero.instNormalizedGCDMonoidCancelCommMonoidWithZero.{u1} G₀ _inst_1 (fun (a : G₀) (b : G₀) => _inst_2 a b)))) a) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) a) 1 (One.toOfNat1.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) a) (InvOneClass.toOne.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) a) (DivInvOneMonoid.toInvOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) a) (DivisionMonoid.toDivInvOneMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) a) (DivisionCommMonoid.toDivisionMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) a) (CommGroupWithZero.toDivisionCommMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G₀) => G₀) a) _inst_1))))))))
Case conversion may be inaccurate. Consider using '#align comm_group_with_zero.normalize_eq_one CommGroupWithZero.normalize_eq_oneₓ'. -/
theorem normalize_eq_one {a : G₀} (h0 : a ≠ 0) : normalize a = 1 := by simp [normalize_apply, h0]
#align comm_group_with_zero.normalize_eq_one CommGroupWithZero.normalize_eq_one

end CommGroupWithZero