# The negative integers ```agda module elementary-number-theory.negative-integers where ``` <details><summary>Imports</summary> ```agda open import elementary-number-theory.integers open import elementary-number-theory.natural-numbers open import elementary-number-theory.nonzero-integers open import foundation.action-on-identifications-functions open import foundation.coproduct-types open import foundation.decidable-subtypes open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.retractions open import foundation.sections open import foundation.sets open import foundation.subtypes open import foundation.transport-along-identifications open import foundation.unit-type open import foundation.universe-levels ``` </details> ## Idea The [integers](elementary-number-theory.integers.md) are defined as a [disjoint sum](foundation-core.coproduct-types.md) of three components. A single element component containing the integer _zero_, and two copies of the [natural numbers](elementary-number-theory.natural-numbers.md), one copy for the _negative integers_ and one copy for the [positive integers](elementary-number-theory.positive-integers.md). Arranged on a number line, we have ```text ⋯ -4 -3 -2 -1 0 1 2 3 4 ⋯ <---+---+---+---] | [---+---+---+---> ``` We say an integer is {{#concept "negative" Disambiguation="integer" Agda=is-negative-ℤ}} if it is an element of the negative component of the integers. ## Definitions ### Negative integers ```agda is-negative-ℤ : ℤ → UU lzero is-negative-ℤ (inl k) = unit is-negative-ℤ (inr k) = empty is-prop-is-negative-ℤ : (x : ℤ) → is-prop (is-negative-ℤ x) is-prop-is-negative-ℤ (inl x) = is-prop-unit is-prop-is-negative-ℤ (inr x) = is-prop-empty subtype-negative-ℤ : subtype lzero ℤ subtype-negative-ℤ x = (is-negative-ℤ x , is-prop-is-negative-ℤ x) negative-ℤ : UU lzero negative-ℤ = type-subtype subtype-negative-ℤ is-negative-eq-ℤ : {x y : ℤ} → x = y → is-negative-ℤ x → is-negative-ℤ y is-negative-eq-ℤ = tr is-negative-ℤ module _ (p : negative-ℤ) where int-negative-ℤ : ℤ int-negative-ℤ = pr1 p is-negative-int-negative-ℤ : is-negative-ℤ int-negative-ℤ is-negative-int-negative-ℤ = pr2 p ``` ### Negative constants ```agda neg-one-negative-ℤ : negative-ℤ neg-one-negative-ℤ = (neg-one-ℤ , star) ``` ## Properties ### Negativity is decidable ```agda is-decidable-is-negative-ℤ : is-decidable-fam is-negative-ℤ is-decidable-is-negative-ℤ (inl x) = inl star is-decidable-is-negative-ℤ (inr x) = inr id decidable-subtype-negative-ℤ : decidable-subtype lzero ℤ decidable-subtype-negative-ℤ x = ( is-negative-ℤ x , is-prop-is-negative-ℤ x , is-decidable-is-negative-ℤ x) ``` ### Negative integers are nonzero ```agda is-nonzero-is-negative-ℤ : {x : ℤ} → is-negative-ℤ x → is-nonzero-ℤ x is-nonzero-is-negative-ℤ {inl x} H () ``` ### The negative integers form a set ```agda is-set-negative-ℤ : is-set negative-ℤ is-set-negative-ℤ = is-set-type-subtype (subtype-negative-ℤ) (is-set-ℤ) ``` ### The predecessor of a negative integer is negative ```agda is-negative-pred-is-negative-ℤ : {x : ℤ} → is-negative-ℤ x → is-negative-ℤ (pred-ℤ x) is-negative-pred-is-negative-ℤ {inl x} H = H pred-negative-ℤ : negative-ℤ → negative-ℤ pred-negative-ℤ (x , H) = (pred-ℤ x , is-negative-pred-is-negative-ℤ H) ``` ### The canonical equivalence between natural numbers and negative integers ```agda negative-int-ℕ : ℕ → negative-ℤ negative-int-ℕ = rec-ℕ neg-one-negative-ℤ (λ _ → pred-negative-ℤ) nat-negative-ℤ : negative-ℤ → ℕ nat-negative-ℤ (inl x , H) = x eq-nat-negative-pred-negative-ℤ : (x : negative-ℤ) → nat-negative-ℤ (pred-negative-ℤ x) = succ-ℕ (nat-negative-ℤ x) eq-nat-negative-pred-negative-ℤ (inl x , H) = refl is-section-nat-negative-ℤ : (x : negative-ℤ) → negative-int-ℕ (nat-negative-ℤ x) = x is-section-nat-negative-ℤ (inl zero-ℕ , H) = refl is-section-nat-negative-ℤ (inl (succ-ℕ x) , H) = ap pred-negative-ℤ (is-section-nat-negative-ℤ (inl x , H)) is-retraction-nat-negative-ℤ : (n : ℕ) → nat-negative-ℤ (negative-int-ℕ n) = n is-retraction-nat-negative-ℤ zero-ℕ = refl is-retraction-nat-negative-ℤ (succ-ℕ n) = eq-nat-negative-pred-negative-ℤ (negative-int-ℕ n) ∙ ap succ-ℕ (is-retraction-nat-negative-ℤ n) is-equiv-negative-int-ℕ : is-equiv negative-int-ℕ pr1 (pr1 is-equiv-negative-int-ℕ) = nat-negative-ℤ pr2 (pr1 is-equiv-negative-int-ℕ) = is-section-nat-negative-ℤ pr1 (pr2 is-equiv-negative-int-ℕ) = nat-negative-ℤ pr2 (pr2 is-equiv-negative-int-ℕ) = is-retraction-nat-negative-ℤ equiv-negative-int-ℕ : ℕ ≃ negative-ℤ pr1 equiv-negative-int-ℕ = negative-int-ℕ pr2 equiv-negative-int-ℕ = is-equiv-negative-int-ℕ ``` ## See also - Relations between negative and positive, nonnegative and nonnpositive integers are derived in [`positive-and-negative-integers`](elementary-number-theory.positive-and-negative-integers.md)