Euclidean domains

Content created by Fredrik Bakke, Egbert Rijke and Fernando Chu.

Created on 2023-04-05.
Last modified on 2024-02-06.

module commutative-algebra.euclidean-domains where
Imports
open import commutative-algebra.commutative-rings
open import commutative-algebra.commutative-semirings
open import commutative-algebra.integral-domains
open import commutative-algebra.trivial-commutative-rings

open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.natural-numbers
open import elementary-number-theory.strict-inequality-natural-numbers

open import foundation.action-on-identifications-binary-functions
open import foundation.action-on-identifications-functions
open import foundation.binary-embeddings
open import foundation.binary-equivalences
open import foundation.cartesian-product-types
open import foundation.coproduct-types
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.equivalences
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.interchange-law
open import foundation.involutions
open import foundation.negation
open import foundation.propositions
open import foundation.sets
open import foundation.unital-binary-operations
open import foundation.universe-levels

open import group-theory.abelian-groups
open import group-theory.commutative-monoids
open import group-theory.groups
open import group-theory.monoids
open import group-theory.semigroups

open import lists.concatenation-lists
open import lists.lists

open import ring-theory.rings
open import ring-theory.semirings

Idea

A Euclidean domain is an integral domain R that has a Euclidean valuation, i.e., a function v : R → ℕ such that for every x y : R, if y is nonzero then there are q r : R with x = q y + r and v r < v y.

Definition

The condition of being a Euclidean valuation

is-euclidean-valuation :
  { l : Level} (R : Integral-Domain l) 
  ( type-Integral-Domain R  ) 
  UU l
is-euclidean-valuation R v =
  ( x y : type-Integral-Domain R) 
  ( is-nonzero-Integral-Domain R y) 
  Σ ( type-Integral-Domain R × type-Integral-Domain R)
    ( λ (q , r) 
      ( Id x (add-Integral-Domain R (mul-Integral-Domain R q y) r)) ×
        ( is-zero-Integral-Domain R r +
        ( v r <-ℕ v y)))

The condition of being a Euclidean domain

is-euclidean-domain-Integral-Domain :
  { l : Level} (R : Integral-Domain l)  UU l
is-euclidean-domain-Integral-Domain R =
  Σ (type-Integral-Domain R  ) (is-euclidean-valuation R)

Euclidean domains

Euclidean-Domain : (l : Level)  UU (lsuc l)
Euclidean-Domain l =
  Σ (Integral-Domain l) is-euclidean-domain-Integral-Domain

module _
  {l : Level} (R : Euclidean-Domain l)
  where

  integral-domain-Euclidean-Domain : Integral-Domain l
  integral-domain-Euclidean-Domain = pr1 R

  is-euclidean-domain-Euclidean-Domain :
    is-euclidean-domain-Integral-Domain integral-domain-Euclidean-Domain
  is-euclidean-domain-Euclidean-Domain = pr2 R

  commutative-ring-Euclidean-Domain : Commutative-Ring l
  commutative-ring-Euclidean-Domain =
    commutative-ring-Integral-Domain integral-domain-Euclidean-Domain

  has-cancellation-property-Euclidean-Domain :
    cancellation-property-Commutative-Ring commutative-ring-Euclidean-Domain
  has-cancellation-property-Euclidean-Domain =
    has-cancellation-property-Integral-Domain
      integral-domain-Euclidean-Domain

  is-nontrivial-Euclidean-Domain :
    is-nontrivial-Commutative-Ring commutative-ring-Euclidean-Domain
  is-nontrivial-Euclidean-Domain =
    is-nontrivial-Integral-Domain
      integral-domain-Euclidean-Domain

  ab-Euclidean-Domain : Ab l
  ab-Euclidean-Domain =
    ab-Commutative-Ring commutative-ring-Euclidean-Domain

  ring-Euclidean-Domain : Ring l
  ring-Euclidean-Domain =
    ring-Integral-Domain integral-domain-Euclidean-Domain

  set-Euclidean-Domain : Set l
  set-Euclidean-Domain = set-Ring ring-Euclidean-Domain

  type-Euclidean-Domain : UU l
  type-Euclidean-Domain = type-Ring ring-Euclidean-Domain

  is-set-type-Euclidean-Domain : is-set type-Euclidean-Domain
  is-set-type-Euclidean-Domain = is-set-type-Ring ring-Euclidean-Domain

Addition in a Euclidean domain

  has-associative-add-Euclidean-Domain :
    has-associative-mul-Set set-Euclidean-Domain
  has-associative-add-Euclidean-Domain =
    has-associative-add-Integral-Domain integral-domain-Euclidean-Domain

  add-Euclidean-Domain :
    type-Euclidean-Domain  type-Euclidean-Domain  type-Euclidean-Domain
  add-Euclidean-Domain = add-Integral-Domain integral-domain-Euclidean-Domain

  add-Euclidean-Domain' :
    type-Euclidean-Domain  type-Euclidean-Domain  type-Euclidean-Domain
  add-Euclidean-Domain' = add-Integral-Domain' integral-domain-Euclidean-Domain

  ap-add-Euclidean-Domain :
    {x x' y y' : type-Euclidean-Domain} 
    (x  x')  (y  y') 
    add-Euclidean-Domain x y  add-Euclidean-Domain x' y'
  ap-add-Euclidean-Domain =
    ap-add-Integral-Domain integral-domain-Euclidean-Domain

  associative-add-Euclidean-Domain :
    (x y z : type-Euclidean-Domain) 
    ( add-Euclidean-Domain (add-Euclidean-Domain x y) z) 
    ( add-Euclidean-Domain x (add-Euclidean-Domain y z))
  associative-add-Euclidean-Domain =
    associative-add-Integral-Domain integral-domain-Euclidean-Domain

  additive-semigroup-Euclidean-Domain : Semigroup l
  additive-semigroup-Euclidean-Domain = semigroup-Ab ab-Euclidean-Domain

  is-group-additive-semigroup-Euclidean-Domain :
    is-group additive-semigroup-Euclidean-Domain
  is-group-additive-semigroup-Euclidean-Domain =
    is-group-Ab ab-Euclidean-Domain

  commutative-add-Euclidean-Domain :
    (x y : type-Euclidean-Domain) 
    Id (add-Euclidean-Domain x y) (add-Euclidean-Domain y x)
  commutative-add-Euclidean-Domain = commutative-add-Ab ab-Euclidean-Domain

  interchange-add-add-Euclidean-Domain :
    (x y x' y' : type-Euclidean-Domain) 
    ( add-Euclidean-Domain
      ( add-Euclidean-Domain x y)
      ( add-Euclidean-Domain x' y')) 
    ( add-Euclidean-Domain
      ( add-Euclidean-Domain x x')
      ( add-Euclidean-Domain y y'))
  interchange-add-add-Euclidean-Domain =
    interchange-add-add-Integral-Domain integral-domain-Euclidean-Domain

  right-swap-add-Euclidean-Domain :
    (x y z : type-Euclidean-Domain) 
    ( add-Euclidean-Domain (add-Euclidean-Domain x y) z) 
    ( add-Euclidean-Domain (add-Euclidean-Domain x z) y)
  right-swap-add-Euclidean-Domain =
    right-swap-add-Integral-Domain integral-domain-Euclidean-Domain

  left-swap-add-Euclidean-Domain :
    (x y z : type-Euclidean-Domain) 
    ( add-Euclidean-Domain x (add-Euclidean-Domain y z)) 
    ( add-Euclidean-Domain y (add-Euclidean-Domain x z))
  left-swap-add-Euclidean-Domain =
    left-swap-add-Integral-Domain integral-domain-Euclidean-Domain

  is-equiv-add-Euclidean-Domain :
    (x : type-Euclidean-Domain)  is-equiv (add-Euclidean-Domain x)
  is-equiv-add-Euclidean-Domain = is-equiv-add-Ab ab-Euclidean-Domain

  is-equiv-add-Euclidean-Domain' :
    (x : type-Euclidean-Domain)  is-equiv (add-Euclidean-Domain' x)
  is-equiv-add-Euclidean-Domain' = is-equiv-add-Ab' ab-Euclidean-Domain

  is-binary-equiv-add-Euclidean-Domain : is-binary-equiv add-Euclidean-Domain
  pr1 is-binary-equiv-add-Euclidean-Domain = is-equiv-add-Euclidean-Domain'
  pr2 is-binary-equiv-add-Euclidean-Domain = is-equiv-add-Euclidean-Domain

  is-binary-emb-add-Euclidean-Domain : is-binary-emb add-Euclidean-Domain
  is-binary-emb-add-Euclidean-Domain = is-binary-emb-add-Ab ab-Euclidean-Domain

  is-emb-add-Euclidean-Domain :
    (x : type-Euclidean-Domain)  is-emb (add-Euclidean-Domain x)
  is-emb-add-Euclidean-Domain = is-emb-add-Ab ab-Euclidean-Domain

  is-emb-add-Euclidean-Domain' :
    (x : type-Euclidean-Domain)  is-emb (add-Euclidean-Domain' x)
  is-emb-add-Euclidean-Domain' = is-emb-add-Ab' ab-Euclidean-Domain

  is-injective-add-Euclidean-Domain :
    (x : type-Euclidean-Domain)  is-injective (add-Euclidean-Domain x)
  is-injective-add-Euclidean-Domain = is-injective-add-Ab ab-Euclidean-Domain

  is-injective-add-Euclidean-Domain' :
    (x : type-Euclidean-Domain)  is-injective (add-Euclidean-Domain' x)
  is-injective-add-Euclidean-Domain' = is-injective-add-Ab' ab-Euclidean-Domain

The zero element of a Euclidean domain

  has-zero-Euclidean-Domain : is-unital add-Euclidean-Domain
  has-zero-Euclidean-Domain =
    has-zero-Integral-Domain integral-domain-Euclidean-Domain

  zero-Euclidean-Domain : type-Euclidean-Domain
  zero-Euclidean-Domain =
    zero-Integral-Domain integral-domain-Euclidean-Domain

  is-zero-Euclidean-Domain : type-Euclidean-Domain  UU l
  is-zero-Euclidean-Domain =
    is-zero-Integral-Domain integral-domain-Euclidean-Domain

  is-nonzero-Euclidean-Domain : type-Euclidean-Domain  UU l
  is-nonzero-Euclidean-Domain =
    is-nonzero-Integral-Domain integral-domain-Euclidean-Domain

  is-zero-euclidean-domain-Prop : type-Euclidean-Domain  Prop l
  is-zero-euclidean-domain-Prop x =
    Id-Prop set-Euclidean-Domain x zero-Euclidean-Domain

  is-nonzero-euclidean-domain-Prop : type-Euclidean-Domain  Prop l
  is-nonzero-euclidean-domain-Prop x =
    neg-Prop (is-zero-euclidean-domain-Prop x)

  left-unit-law-add-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    add-Euclidean-Domain zero-Euclidean-Domain x  x
  left-unit-law-add-Euclidean-Domain =
    left-unit-law-add-Integral-Domain integral-domain-Euclidean-Domain

  right-unit-law-add-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    add-Euclidean-Domain x zero-Euclidean-Domain  x
  right-unit-law-add-Euclidean-Domain =
    right-unit-law-add-Integral-Domain integral-domain-Euclidean-Domain

Additive inverses in a Euclidean domain

  has-negatives-Euclidean-Domain :
    is-group' additive-semigroup-Euclidean-Domain has-zero-Euclidean-Domain
  has-negatives-Euclidean-Domain = has-negatives-Ab ab-Euclidean-Domain

  neg-Euclidean-Domain : type-Euclidean-Domain  type-Euclidean-Domain
  neg-Euclidean-Domain = neg-Integral-Domain integral-domain-Euclidean-Domain

  left-inverse-law-add-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    add-Euclidean-Domain (neg-Euclidean-Domain x) x  zero-Euclidean-Domain
  left-inverse-law-add-Euclidean-Domain =
    left-inverse-law-add-Integral-Domain integral-domain-Euclidean-Domain

  right-inverse-law-add-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    add-Euclidean-Domain x (neg-Euclidean-Domain x)  zero-Euclidean-Domain
  right-inverse-law-add-Euclidean-Domain =
    right-inverse-law-add-Integral-Domain integral-domain-Euclidean-Domain

  neg-neg-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    neg-Euclidean-Domain (neg-Euclidean-Domain x)  x
  neg-neg-Euclidean-Domain = neg-neg-Ab ab-Euclidean-Domain

  distributive-neg-add-Euclidean-Domain :
    (x y : type-Euclidean-Domain) 
    neg-Euclidean-Domain (add-Euclidean-Domain x y) 
    add-Euclidean-Domain (neg-Euclidean-Domain x) (neg-Euclidean-Domain y)
  distributive-neg-add-Euclidean-Domain =
    distributive-neg-add-Ab ab-Euclidean-Domain

Multiplication in a Euclidean domain

  has-associative-mul-Euclidean-Domain :
    has-associative-mul-Set set-Euclidean-Domain
  has-associative-mul-Euclidean-Domain =
    has-associative-mul-Integral-Domain integral-domain-Euclidean-Domain

  mul-Euclidean-Domain :
    (x y : type-Euclidean-Domain)  type-Euclidean-Domain
  mul-Euclidean-Domain =
    mul-Integral-Domain integral-domain-Euclidean-Domain

  mul-Euclidean-Domain' :
    (x y : type-Euclidean-Domain)  type-Euclidean-Domain
  mul-Euclidean-Domain' =
    mul-Integral-Domain' integral-domain-Euclidean-Domain

  ap-mul-Euclidean-Domain :
    {x x' y y' : type-Euclidean-Domain} (p : Id x x') (q : Id y y') 
    Id (mul-Euclidean-Domain x y) (mul-Euclidean-Domain x' y')
  ap-mul-Euclidean-Domain p q = ap-binary mul-Euclidean-Domain p q

  associative-mul-Euclidean-Domain :
    (x y z : type-Euclidean-Domain) 
    mul-Euclidean-Domain (mul-Euclidean-Domain x y) z 
    mul-Euclidean-Domain x (mul-Euclidean-Domain y z)
  associative-mul-Euclidean-Domain =
    associative-mul-Integral-Domain integral-domain-Euclidean-Domain

  multiplicative-semigroup-Euclidean-Domain : Semigroup l
  multiplicative-semigroup-Euclidean-Domain =
    multiplicative-semigroup-Integral-Domain
      integral-domain-Euclidean-Domain

  left-distributive-mul-add-Euclidean-Domain :
    (x y z : type-Euclidean-Domain) 
    ( mul-Euclidean-Domain x (add-Euclidean-Domain y z)) 
    ( add-Euclidean-Domain
      ( mul-Euclidean-Domain x y)
      ( mul-Euclidean-Domain x z))
  left-distributive-mul-add-Euclidean-Domain =
    left-distributive-mul-add-Integral-Domain
      integral-domain-Euclidean-Domain

  right-distributive-mul-add-Euclidean-Domain :
    (x y z : type-Euclidean-Domain) 
    ( mul-Euclidean-Domain (add-Euclidean-Domain x y) z) 
    ( add-Euclidean-Domain
      ( mul-Euclidean-Domain x z)
      ( mul-Euclidean-Domain y z))
  right-distributive-mul-add-Euclidean-Domain =
    right-distributive-mul-add-Integral-Domain
      integral-domain-Euclidean-Domain

  commutative-mul-Euclidean-Domain :
    (x y : type-Euclidean-Domain) 
    mul-Euclidean-Domain x y  mul-Euclidean-Domain y x
  commutative-mul-Euclidean-Domain =
    commutative-mul-Integral-Domain
      integral-domain-Euclidean-Domain

Multiplicative units in a Euclidean domain

  is-unital-Euclidean-Domain : is-unital mul-Euclidean-Domain
  is-unital-Euclidean-Domain =
    is-unital-Integral-Domain
      integral-domain-Euclidean-Domain

  multiplicative-monoid-Euclidean-Domain : Monoid l
  multiplicative-monoid-Euclidean-Domain =
    multiplicative-monoid-Integral-Domain
      integral-domain-Euclidean-Domain

  one-Euclidean-Domain : type-Euclidean-Domain
  one-Euclidean-Domain =
    one-Integral-Domain
      integral-domain-Euclidean-Domain

  left-unit-law-mul-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    mul-Euclidean-Domain one-Euclidean-Domain x  x
  left-unit-law-mul-Euclidean-Domain =
    left-unit-law-mul-Integral-Domain
      integral-domain-Euclidean-Domain

  right-unit-law-mul-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    mul-Euclidean-Domain x one-Euclidean-Domain  x
  right-unit-law-mul-Euclidean-Domain =
    right-unit-law-mul-Integral-Domain
      integral-domain-Euclidean-Domain

  right-swap-mul-Euclidean-Domain :
    (x y z : type-Euclidean-Domain) 
    mul-Euclidean-Domain (mul-Euclidean-Domain x y) z 
    mul-Euclidean-Domain (mul-Euclidean-Domain x z) y
  right-swap-mul-Euclidean-Domain x y z =
    ( associative-mul-Euclidean-Domain x y z) 
    ( ( ap
        ( mul-Euclidean-Domain x)
        ( commutative-mul-Euclidean-Domain y z)) 
      ( inv (associative-mul-Euclidean-Domain x z y)))

  left-swap-mul-Euclidean-Domain :
    (x y z : type-Euclidean-Domain) 
    mul-Euclidean-Domain x (mul-Euclidean-Domain y z) 
    mul-Euclidean-Domain y (mul-Euclidean-Domain x z)
  left-swap-mul-Euclidean-Domain x y z =
    ( inv (associative-mul-Euclidean-Domain x y z)) 
    ( ( ap
        ( mul-Euclidean-Domain' z)
        ( commutative-mul-Euclidean-Domain x y)) 
      ( associative-mul-Euclidean-Domain y x z))

  interchange-mul-mul-Euclidean-Domain :
    (x y z w : type-Euclidean-Domain) 
    mul-Euclidean-Domain
      ( mul-Euclidean-Domain x y)
      ( mul-Euclidean-Domain z w) 
    mul-Euclidean-Domain
      ( mul-Euclidean-Domain x z)
      ( mul-Euclidean-Domain y w)
  interchange-mul-mul-Euclidean-Domain =
    interchange-law-commutative-and-associative
      mul-Euclidean-Domain
      commutative-mul-Euclidean-Domain
      associative-mul-Euclidean-Domain

The zero laws for multiplication of a Euclidean domain

  left-zero-law-mul-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    mul-Euclidean-Domain zero-Euclidean-Domain x 
    zero-Euclidean-Domain
  left-zero-law-mul-Euclidean-Domain =
    left-zero-law-mul-Integral-Domain integral-domain-Euclidean-Domain

  right-zero-law-mul-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    mul-Euclidean-Domain x zero-Euclidean-Domain 
    zero-Euclidean-Domain
  right-zero-law-mul-Euclidean-Domain =
    right-zero-law-mul-Integral-Domain integral-domain-Euclidean-Domain

Euclidean domains are commutative semirings

  multiplicative-commutative-monoid-Euclidean-Domain : Commutative-Monoid l
  multiplicative-commutative-monoid-Euclidean-Domain =
    multiplicative-commutative-monoid-Integral-Domain
      integral-domain-Euclidean-Domain

  semiring-Euclidean-Domain : Semiring l
  semiring-Euclidean-Domain =
    semiring-Integral-Domain integral-domain-Euclidean-Domain

  commutative-semiring-Euclidean-Domain : Commutative-Semiring l
  commutative-semiring-Euclidean-Domain =
    commutative-semiring-Integral-Domain
      integral-domain-Euclidean-Domain

Computing multiplication with minus one in a Euclidean domain

  neg-one-Euclidean-Domain : type-Euclidean-Domain
  neg-one-Euclidean-Domain =
    neg-one-Integral-Domain
      integral-domain-Euclidean-Domain

  mul-neg-one-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    mul-Euclidean-Domain neg-one-Euclidean-Domain x 
    neg-Euclidean-Domain x
  mul-neg-one-Euclidean-Domain =
    mul-neg-one-Integral-Domain
      integral-domain-Euclidean-Domain

  mul-neg-one-Euclidean-Domain' :
    (x : type-Euclidean-Domain) 
    mul-Euclidean-Domain x neg-one-Euclidean-Domain 
    neg-Euclidean-Domain x
  mul-neg-one-Euclidean-Domain' =
    mul-neg-one-Integral-Domain'
      integral-domain-Euclidean-Domain

  is-involution-mul-neg-one-Euclidean-Domain :
    is-involution (mul-Euclidean-Domain neg-one-Euclidean-Domain)
  is-involution-mul-neg-one-Euclidean-Domain =
    is-involution-mul-neg-one-Integral-Domain
      integral-domain-Euclidean-Domain

  is-involution-mul-neg-one-Euclidean-Domain' :
    is-involution (mul-Euclidean-Domain' neg-one-Euclidean-Domain)
  is-involution-mul-neg-one-Euclidean-Domain' =
    is-involution-mul-neg-one-Integral-Domain'
      integral-domain-Euclidean-Domain

Left and right negative laws for multiplication

  left-negative-law-mul-Euclidean-Domain :
    (x y : type-Euclidean-Domain) 
    mul-Euclidean-Domain (neg-Euclidean-Domain x) y 
    neg-Euclidean-Domain (mul-Euclidean-Domain x y)
  left-negative-law-mul-Euclidean-Domain =
    left-negative-law-mul-Integral-Domain
      integral-domain-Euclidean-Domain

  right-negative-law-mul-Euclidean-Domain :
    (x y : type-Euclidean-Domain) 
    mul-Euclidean-Domain x (neg-Euclidean-Domain y) 
    neg-Euclidean-Domain (mul-Euclidean-Domain x y)
  right-negative-law-mul-Euclidean-Domain =
    right-negative-law-mul-Integral-Domain
      integral-domain-Euclidean-Domain

  mul-neg-Euclidean-Domain :
    (x y : type-Euclidean-Domain) 
    mul-Euclidean-Domain (neg-Euclidean-Domain x) (neg-Euclidean-Domain y) 
    mul-Euclidean-Domain x y
  mul-neg-Euclidean-Domain =
    mul-neg-Integral-Domain
      integral-domain-Euclidean-Domain

Scalar multiplication of elements of a Euclidean domain by natural numbers

  mul-nat-scalar-Euclidean-Domain :
      type-Euclidean-Domain  type-Euclidean-Domain
  mul-nat-scalar-Euclidean-Domain =
    mul-nat-scalar-Integral-Domain
      integral-domain-Euclidean-Domain

  ap-mul-nat-scalar-Euclidean-Domain :
    {m n : } {x y : type-Euclidean-Domain} 
    (m  n)  (x  y) 
    mul-nat-scalar-Euclidean-Domain m x 
    mul-nat-scalar-Euclidean-Domain n y
  ap-mul-nat-scalar-Euclidean-Domain =
    ap-mul-nat-scalar-Integral-Domain
      integral-domain-Euclidean-Domain

  left-zero-law-mul-nat-scalar-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    mul-nat-scalar-Euclidean-Domain 0 x  zero-Euclidean-Domain
  left-zero-law-mul-nat-scalar-Euclidean-Domain =
    left-zero-law-mul-nat-scalar-Integral-Domain
      integral-domain-Euclidean-Domain

  right-zero-law-mul-nat-scalar-Euclidean-Domain :
    (n : ) 
    mul-nat-scalar-Euclidean-Domain n zero-Euclidean-Domain 
    zero-Euclidean-Domain
  right-zero-law-mul-nat-scalar-Euclidean-Domain =
    right-zero-law-mul-nat-scalar-Integral-Domain
      integral-domain-Euclidean-Domain

  left-unit-law-mul-nat-scalar-Euclidean-Domain :
    (x : type-Euclidean-Domain) 
    mul-nat-scalar-Euclidean-Domain 1 x  x
  left-unit-law-mul-nat-scalar-Euclidean-Domain =
    left-unit-law-mul-nat-scalar-Integral-Domain
      integral-domain-Euclidean-Domain

  left-nat-scalar-law-mul-Euclidean-Domain :
    (n : ) (x y : type-Euclidean-Domain) 
    mul-Euclidean-Domain (mul-nat-scalar-Euclidean-Domain n x) y 
    mul-nat-scalar-Euclidean-Domain n (mul-Euclidean-Domain x y)
  left-nat-scalar-law-mul-Euclidean-Domain =
    left-nat-scalar-law-mul-Integral-Domain
      integral-domain-Euclidean-Domain

  right-nat-scalar-law-mul-Euclidean-Domain :
    (n : ) (x y : type-Euclidean-Domain) 
    mul-Euclidean-Domain x (mul-nat-scalar-Euclidean-Domain n y) 
    mul-nat-scalar-Euclidean-Domain n (mul-Euclidean-Domain x y)
  right-nat-scalar-law-mul-Euclidean-Domain =
    right-nat-scalar-law-mul-Integral-Domain
      integral-domain-Euclidean-Domain

  left-distributive-mul-nat-scalar-add-Euclidean-Domain :
    (n : ) (x y : type-Euclidean-Domain) 
    mul-nat-scalar-Euclidean-Domain n (add-Euclidean-Domain x y) 
    add-Euclidean-Domain
      ( mul-nat-scalar-Euclidean-Domain n x)
      ( mul-nat-scalar-Euclidean-Domain n y)
  left-distributive-mul-nat-scalar-add-Euclidean-Domain =
    left-distributive-mul-nat-scalar-add-Integral-Domain
      integral-domain-Euclidean-Domain

  right-distributive-mul-nat-scalar-add-Euclidean-Domain :
    (m n : ) (x : type-Euclidean-Domain) 
    mul-nat-scalar-Euclidean-Domain (m +ℕ n) x 
    add-Euclidean-Domain
      ( mul-nat-scalar-Euclidean-Domain m x)
      ( mul-nat-scalar-Euclidean-Domain n x)
  right-distributive-mul-nat-scalar-add-Euclidean-Domain =
    right-distributive-mul-nat-scalar-add-Integral-Domain
      integral-domain-Euclidean-Domain

Addition of a list of elements in a Euclidean domain

  add-list-Euclidean-Domain :
    list type-Euclidean-Domain  type-Euclidean-Domain
  add-list-Euclidean-Domain =
    add-list-Integral-Domain integral-domain-Euclidean-Domain

  preserves-concat-add-list-Euclidean-Domain :
    (l1 l2 : list type-Euclidean-Domain) 
    Id
      ( add-list-Euclidean-Domain (concat-list l1 l2))
      ( add-Euclidean-Domain
        ( add-list-Euclidean-Domain l1)
        ( add-list-Euclidean-Domain l2))
  preserves-concat-add-list-Euclidean-Domain =
    preserves-concat-add-list-Integral-Domain
      integral-domain-Euclidean-Domain

Euclidean division in a Euclidean domain

  euclidean-valuation-Euclidean-Domain :
    type-Euclidean-Domain  
  euclidean-valuation-Euclidean-Domain =
    pr1 is-euclidean-domain-Euclidean-Domain

  euclidean-division-Euclidean-Domain :
    ( x y : type-Euclidean-Domain) 
    ( is-nonzero-Euclidean-Domain y) 
    type-Euclidean-Domain × type-Euclidean-Domain
  euclidean-division-Euclidean-Domain x y p =
    pr1 (pr2 is-euclidean-domain-Euclidean-Domain x y p)

  quotient-euclidean-division-Euclidean-Domain :
    ( x y : type-Euclidean-Domain) 
    ( is-nonzero-Euclidean-Domain y) 
    type-Euclidean-Domain
  quotient-euclidean-division-Euclidean-Domain x y p =
    pr1 (euclidean-division-Euclidean-Domain x y p)

  remainder-euclidean-division-Euclidean-Domain :
    ( x y : type-Euclidean-Domain) 
    ( is-nonzero-Euclidean-Domain y) 
    type-Euclidean-Domain
  remainder-euclidean-division-Euclidean-Domain x y p =
    pr2 (euclidean-division-Euclidean-Domain x y p)

  equation-euclidean-division-Euclidean-Domain :
    ( x y : type-Euclidean-Domain) 
    ( p : is-nonzero-Euclidean-Domain y) 
    ( Id
      ( x)
      ( add-Euclidean-Domain
        ( mul-Euclidean-Domain
          ( quotient-euclidean-division-Euclidean-Domain x y p)
          ( y))
        ( remainder-euclidean-division-Euclidean-Domain x y p)))
  equation-euclidean-division-Euclidean-Domain x y p =
    pr1 (pr2 (pr2 is-euclidean-domain-Euclidean-Domain x y p))

  remainder-condition-euclidean-division-Euclidean-Domain :
    ( x y : type-Euclidean-Domain) 
    ( p : is-nonzero-Integral-Domain integral-domain-Euclidean-Domain y) 
    ( is-zero-Euclidean-Domain
      ( remainder-euclidean-division-Euclidean-Domain x y p)) +
    ( euclidean-valuation-Euclidean-Domain
      ( remainder-euclidean-division-Euclidean-Domain x y p) <-ℕ
    ( euclidean-valuation-Euclidean-Domain y))
  remainder-condition-euclidean-division-Euclidean-Domain x y p =
    pr2 (pr2 (pr2 is-euclidean-domain-Euclidean-Domain x y p))

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