The integers

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Fernando Chu, Bryan Lu and malarbol.

Created on 2022-01-26.
Last modified on 2024-10-16.

module elementary-number-theory.integers where
Imports 
open import elementary-number-theory.equality-natural-numbers
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.coproduct-types
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.empty-types
open import foundation.equality-coproduct-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.negated-equality
open import foundation.negation
open import foundation.propositions
open import foundation.retractions
open import foundation.sections
open import foundation.sets
open import foundation.unit-type
open import foundation.universe-levels

open import structured-types.types-equipped-with-endomorphisms

Idea

The type of integers is an extension of the type of natural numbers including negative whole numbers.

Definitions

The type of integers

ℤ : UU lzero
ℤ = ℕ + (unit + ℕ)

{-# BUILTIN INTEGER ℤ #-}

Inclusion of the negative integers

in-neg-ℤ : ℕ → ℤ
in-neg-ℤ n = inl n

neg-one-ℤ : ℤ
neg-one-ℤ = in-neg-ℤ zero-ℕ

is-neg-one-ℤ : ℤ → UU lzero
is-neg-one-ℤ x = (x ＝ neg-one-ℤ)

Zero

zero-ℤ : ℤ
zero-ℤ = inr (inl star)

is-zero-ℤ : ℤ → UU lzero
is-zero-ℤ x = (x ＝ zero-ℤ)

eq-is-zero-ℤ : {a b : ℤ} → is-zero-ℤ a → is-zero-ℤ b → a ＝ b
eq-is-zero-ℤ {a} {b} H K = H ∙ inv K

Inclusion of the positive integers

in-pos-ℤ : ℕ → ℤ
in-pos-ℤ n = inr (inr n)

one-ℤ : ℤ
one-ℤ = in-pos-ℤ zero-ℕ

is-one-ℤ : ℤ → UU lzero
is-one-ℤ x = (x ＝ one-ℤ)

Inclusion of the natural numbers

int-ℕ : ℕ → ℤ
int-ℕ zero-ℕ = zero-ℤ
int-ℕ (succ-ℕ n) = in-pos-ℤ n

is-injective-int-ℕ : is-injective int-ℕ
is-injective-int-ℕ {zero-ℕ} {zero-ℕ} refl = refl
is-injective-int-ℕ {succ-ℕ x} {succ-ℕ y} refl = refl

Induction principle on the type of integers

ind-ℤ :
  {l : Level} (P : ℤ → UU l) →
  P neg-one-ℤ → ((n : ℕ) → P (inl n) → P (inl (succ-ℕ n))) →
  P zero-ℤ →
  P one-ℤ → ((n : ℕ) → P (inr (inr (n))) → P (inr (inr (succ-ℕ n)))) →
  (k : ℤ) → P k
ind-ℤ P p-1 p-S p0 p1 pS (inl zero-ℕ) = p-1
ind-ℤ P p-1 p-S p0 p1 pS (inl (succ-ℕ x)) =
  p-S x (ind-ℤ P p-1 p-S p0 p1 pS (inl x))
ind-ℤ P p-1 p-S p0 p1 pS (inr (inl star)) = p0
ind-ℤ P p-1 p-S p0 p1 pS (inr (inr zero-ℕ)) = p1
ind-ℤ P p-1 p-S p0 p1 pS (inr (inr (succ-ℕ x))) =
  pS x (ind-ℤ P p-1 p-S p0 p1 pS (inr (inr (x))))

The successor and predecessor functions on ℤ

succ-ℤ : ℤ → ℤ
succ-ℤ (inl zero-ℕ) = zero-ℤ
succ-ℤ (inl (succ-ℕ x)) = inl x
succ-ℤ (inr (inl star)) = one-ℤ
succ-ℤ (inr (inr x)) = inr (inr (succ-ℕ x))

pred-ℤ : ℤ → ℤ
pred-ℤ (inl x) = inl (succ-ℕ x)
pred-ℤ (inr (inl star)) = inl zero-ℕ
pred-ℤ (inr (inr zero-ℕ)) = inr (inl star)
pred-ℤ (inr (inr (succ-ℕ x))) = inr (inr x)

ℤ-Type-With-Endomorphism : Type-With-Endomorphism lzero
pr1 ℤ-Type-With-Endomorphism = ℤ
pr2 ℤ-Type-With-Endomorphism = succ-ℤ

The negative of an integer

neg-ℤ : ℤ → ℤ
neg-ℤ (inl x) = inr (inr x)
neg-ℤ (inr (inl star)) = inr (inl star)
neg-ℤ (inr (inr x)) = inl x

Properties

The type of integers is a set

abstract
  is-set-ℤ : is-set ℤ
  is-set-ℤ = is-set-coproduct is-set-ℕ (is-set-coproduct is-set-unit is-set-ℕ)

ℤ-Set : Set lzero
pr1 ℤ-Set = ℤ
pr2 ℤ-Set = is-set-ℤ

The successor and predecessor functions on ℤ are mutual inverses

abstract
  is-retraction-pred-ℤ : is-retraction succ-ℤ pred-ℤ
  is-retraction-pred-ℤ (inl zero-ℕ) = refl
  is-retraction-pred-ℤ (inl (succ-ℕ x)) = refl
  is-retraction-pred-ℤ (inr (inl _)) = refl
  is-retraction-pred-ℤ (inr (inr zero-ℕ)) = refl
  is-retraction-pred-ℤ (inr (inr (succ-ℕ x))) = refl

  is-section-pred-ℤ : is-section succ-ℤ pred-ℤ
  is-section-pred-ℤ (inl zero-ℕ) = refl
  is-section-pred-ℤ (inl (succ-ℕ x)) = refl
  is-section-pred-ℤ (inr (inl _)) = refl
  is-section-pred-ℤ (inr (inr zero-ℕ)) = refl
  is-section-pred-ℤ (inr (inr (succ-ℕ x))) = refl

abstract
  is-equiv-succ-ℤ : is-equiv succ-ℤ
  pr1 (pr1 is-equiv-succ-ℤ) = pred-ℤ
  pr2 (pr1 is-equiv-succ-ℤ) = is-section-pred-ℤ
  pr1 (pr2 is-equiv-succ-ℤ) = pred-ℤ
  pr2 (pr2 is-equiv-succ-ℤ) = is-retraction-pred-ℤ

equiv-succ-ℤ : ℤ ≃ ℤ
pr1 equiv-succ-ℤ = succ-ℤ
pr2 equiv-succ-ℤ = is-equiv-succ-ℤ

abstract
  is-equiv-pred-ℤ : is-equiv pred-ℤ
  pr1 (pr1 is-equiv-pred-ℤ) = succ-ℤ
  pr2 (pr1 is-equiv-pred-ℤ) = is-retraction-pred-ℤ
  pr1 (pr2 is-equiv-pred-ℤ) = succ-ℤ
  pr2 (pr2 is-equiv-pred-ℤ) = is-section-pred-ℤ

equiv-pred-ℤ : ℤ ≃ ℤ
pr1 equiv-pred-ℤ = pred-ℤ
pr2 equiv-pred-ℤ = is-equiv-pred-ℤ

The successor function on ℤ is injective and has no fixed points

abstract
  is-injective-succ-ℤ : is-injective succ-ℤ
  is-injective-succ-ℤ {x} {y} p =
    inv (is-retraction-pred-ℤ x) ∙ ap pred-ℤ p ∙ is-retraction-pred-ℤ y

  has-no-fixed-points-succ-ℤ : (x : ℤ) → succ-ℤ x ≠ x
  has-no-fixed-points-succ-ℤ (inl zero-ℕ) ()
  has-no-fixed-points-succ-ℤ (inl (succ-ℕ x)) ()
  has-no-fixed-points-succ-ℤ (inr (inl star)) ()

The negative function is an involution

abstract
  neg-neg-ℤ : neg-ℤ ∘ neg-ℤ ~ id
  neg-neg-ℤ (inl n) = refl
  neg-neg-ℤ (inr (inl star)) = refl
  neg-neg-ℤ (inr (inr n)) = refl

abstract
  is-equiv-neg-ℤ : is-equiv neg-ℤ
  pr1 (pr1 is-equiv-neg-ℤ) = neg-ℤ
  pr2 (pr1 is-equiv-neg-ℤ) = neg-neg-ℤ
  pr1 (pr2 is-equiv-neg-ℤ) = neg-ℤ
  pr2 (pr2 is-equiv-neg-ℤ) = neg-neg-ℤ

equiv-neg-ℤ : ℤ ≃ ℤ
pr1 equiv-neg-ℤ = neg-ℤ
pr2 equiv-neg-ℤ = is-equiv-neg-ℤ

abstract
  is-emb-neg-ℤ : is-emb neg-ℤ
  is-emb-neg-ℤ = is-emb-is-equiv is-equiv-neg-ℤ

emb-neg-ℤ : ℤ ↪ ℤ
pr1 emb-neg-ℤ = neg-ℤ
pr2 emb-neg-ℤ = is-emb-neg-ℤ

abstract
  neg-pred-ℤ : (k : ℤ) → neg-ℤ (pred-ℤ k) ＝ succ-ℤ (neg-ℤ k)
  neg-pred-ℤ (inl x) = refl
  neg-pred-ℤ (inr (inl star)) = refl
  neg-pred-ℤ (inr (inr zero-ℕ)) = refl
  neg-pred-ℤ (inr (inr (succ-ℕ x))) = refl

  neg-succ-ℤ : (x : ℤ) → neg-ℤ (succ-ℤ x) ＝ pred-ℤ (neg-ℤ x)
  neg-succ-ℤ (inl zero-ℕ) = refl
  neg-succ-ℤ (inl (succ-ℕ x)) = refl
  neg-succ-ℤ (inr (inl star)) = refl
  neg-succ-ℤ (inr (inr x)) = refl

  pred-neg-ℤ :
    (k : ℤ) → pred-ℤ (neg-ℤ k) ＝ neg-ℤ (succ-ℤ k)
  pred-neg-ℤ (inl zero-ℕ) = refl
  pred-neg-ℤ (inl (succ-ℕ x)) = refl
  pred-neg-ℤ (inr (inl star)) = refl
  pred-neg-ℤ (inr (inr x)) = refl

The negative function is injective

abstract
  is-injective-neg-ℤ : is-injective neg-ℤ
  is-injective-neg-ℤ {x} {y} p = inv (neg-neg-ℤ x) ∙ ap neg-ℤ p ∙ neg-neg-ℤ y

The integer successor of a natural number is the successor of the natural number

abstract
  succ-int-ℕ : (x : ℕ) → succ-ℤ (int-ℕ x) ＝ int-ℕ (succ-ℕ x)
  succ-int-ℕ zero-ℕ = refl
  succ-int-ℕ (succ-ℕ x) = refl

An integer is zero if its negative is zero

abstract
  is-zero-is-zero-neg-ℤ : (x : ℤ) → is-zero-ℤ (neg-ℤ x) → is-zero-ℤ x
  is-zero-is-zero-neg-ℤ (inr (inl star)) H = refl

See also

Recent changes