Addition on the integers

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, malarbol and Victor Blanchi.

Created on 2022-01-26.
Last modified on 2024-04-11.

module elementary-number-theory.addition-integers where
open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.integers
open import elementary-number-theory.natural-numbers
open import elementary-number-theory.nonnegative-integers
open import elementary-number-theory.positive-and-negative-integers
open import elementary-number-theory.positive-integers

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.coproduct-types
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.equivalences
open import foundation.function-types
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.interchange-law
open import foundation.sets
open import foundation.unit-type


We introduce addition on the integers and derive its basic properties with respect to succ-ℤ and neg-ℤ.


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

add-ℤ' :     
add-ℤ' x y = add-ℤ y x

infixl 35 _+ℤ_
_+ℤ_ = add-ℤ

ap-add-ℤ :
  {x y x' y' : }  x  x'  y  y'  x +ℤ y  x' +ℤ y'
ap-add-ℤ p q = ap-binary add-ℤ p q


Unit laws

  left-unit-law-add-ℤ : (k : )  zero-ℤ +ℤ k  k
  left-unit-law-add-ℤ k = refl

  right-unit-law-add-ℤ : (k : )  k +ℤ zero-ℤ  k
  right-unit-law-add-ℤ (inl zero-ℕ) = refl
  right-unit-law-add-ℤ (inl (succ-ℕ x)) =
    ap pred-ℤ (right-unit-law-add-ℤ (inl x))
  right-unit-law-add-ℤ (inr (inl star)) = refl
  right-unit-law-add-ℤ (inr (inr zero-ℕ)) = refl
  right-unit-law-add-ℤ (inr (inr (succ-ℕ x))) =
    ap succ-ℤ (right-unit-law-add-ℤ (inr (inr x)))

Left and right predecessor laws

  left-predecessor-law-add-ℤ :
    (x y : )  pred-ℤ x +ℤ y  pred-ℤ (x +ℤ y)
  left-predecessor-law-add-ℤ (inl n) y = refl
  left-predecessor-law-add-ℤ (inr (inl star)) y = refl
  left-predecessor-law-add-ℤ (inr (inr zero-ℕ)) y =
    inv (is-retraction-pred-ℤ y)
  left-predecessor-law-add-ℤ (inr (inr (succ-ℕ x))) y =
    inv (is-retraction-pred-ℤ ((inr (inr x)) +ℤ y))

  right-predecessor-law-add-ℤ :
    (x y : )  x +ℤ pred-ℤ y  pred-ℤ (x +ℤ y)
  right-predecessor-law-add-ℤ (inl zero-ℕ) n = refl
  right-predecessor-law-add-ℤ (inl (succ-ℕ m)) n =
    ap pred-ℤ (right-predecessor-law-add-ℤ (inl m) n)
  right-predecessor-law-add-ℤ (inr (inl star)) n = refl
  right-predecessor-law-add-ℤ (inr (inr zero-ℕ)) n =
      succ-ℤ (pred-ℤ n)
        by is-section-pred-ℤ n
       pred-ℤ (succ-ℤ n)
        by inv (is-retraction-pred-ℤ n)
  right-predecessor-law-add-ℤ (inr (inr (succ-ℕ x))) n =
      succ-ℤ (inr (inr x) +ℤ pred-ℤ n)
       succ-ℤ (pred-ℤ (inr (inr x) +ℤ n))
        by ap succ-ℤ (right-predecessor-law-add-ℤ (inr (inr x)) n)
       inr (inr x) +ℤ n
        by is-section-pred-ℤ ((inr (inr x)) +ℤ n)
       pred-ℤ (succ-ℤ (inr (inr x) +ℤ n))
        by inv (is-retraction-pred-ℤ ((inr (inr x)) +ℤ n))

Left and right successor laws

  left-successor-law-add-ℤ :
    (x y : )  succ-ℤ x +ℤ y  succ-ℤ (x +ℤ y)
  left-successor-law-add-ℤ (inl zero-ℕ) y =
    inv (is-section-pred-ℤ y)
  left-successor-law-add-ℤ (inl (succ-ℕ x)) y =
      inl x +ℤ y
       succ-ℤ (pred-ℤ (inl x +ℤ y))
        by inv (is-section-pred-ℤ ((inl x) +ℤ y))
       succ-ℤ (pred-ℤ (inl x) +ℤ y)
        by ap succ-ℤ (inv (left-predecessor-law-add-ℤ (inl x) y))
  left-successor-law-add-ℤ (inr (inl star)) y = refl
  left-successor-law-add-ℤ (inr (inr x)) y = refl

  right-successor-law-add-ℤ :
    (x y : )  x +ℤ succ-ℤ y  succ-ℤ (x +ℤ y)
  right-successor-law-add-ℤ (inl zero-ℕ) y =
      pred-ℤ (succ-ℤ y)
        by is-retraction-pred-ℤ y
       succ-ℤ (pred-ℤ y)
        by inv (is-section-pred-ℤ y)
  right-successor-law-add-ℤ (inl (succ-ℕ x)) y =
      pred-ℤ (inl x +ℤ succ-ℤ y)
       pred-ℤ (succ-ℤ (inl x +ℤ y))
        by ap pred-ℤ (right-successor-law-add-ℤ (inl x) y)
       inl x +ℤ y
        by is-retraction-pred-ℤ ((inl x) +ℤ y)
       succ-ℤ (pred-ℤ (inl x +ℤ y))
        by inv (is-section-pred-ℤ ((inl x) +ℤ y))
  right-successor-law-add-ℤ (inr (inl star)) y = refl
  right-successor-law-add-ℤ (inr (inr zero-ℕ)) y = refl
  right-successor-law-add-ℤ (inr (inr (succ-ℕ x))) y =
    ap succ-ℤ (right-successor-law-add-ℤ (inr (inr x)) y)

The successor of an integer is that integer plus one

  is-right-add-one-succ-ℤ : (x : )  succ-ℤ x  x +ℤ one-ℤ
  is-right-add-one-succ-ℤ x =
      succ-ℤ x
       succ-ℤ (x +ℤ zero-ℤ)
        by inv (ap succ-ℤ (right-unit-law-add-ℤ x))
       x +ℤ one-ℤ
        by inv (right-successor-law-add-ℤ x zero-ℤ)

  is-left-add-one-succ-ℤ : (x : )  succ-ℤ x  one-ℤ +ℤ x
  is-left-add-one-succ-ℤ x = inv (left-successor-law-add-ℤ zero-ℤ x)

  left-add-one-ℤ : (x : )  one-ℤ +ℤ x  succ-ℤ x
  left-add-one-ℤ x = refl

  right-add-one-ℤ : (x : )  x +ℤ one-ℤ  succ-ℤ x
  right-add-one-ℤ x = inv (is-right-add-one-succ-ℤ x)

The predecessor of an integer is that integer minus one

  is-left-add-neg-one-pred-ℤ : (x : )  pred-ℤ x  neg-one-ℤ +ℤ x
  is-left-add-neg-one-pred-ℤ x =
    inv (left-predecessor-law-add-ℤ zero-ℤ x)

  is-right-add-neg-one-pred-ℤ : (x : )  pred-ℤ x  x +ℤ neg-one-ℤ
  is-right-add-neg-one-pred-ℤ x =
      pred-ℤ x
       pred-ℤ (x +ℤ zero-ℤ)
        by inv (ap pred-ℤ (right-unit-law-add-ℤ x))
       x +ℤ neg-one-ℤ
        by inv (right-predecessor-law-add-ℤ x zero-ℤ)

  left-add-neg-one-ℤ : (x : )  neg-one-ℤ +ℤ x  pred-ℤ x
  left-add-neg-one-ℤ x = refl

  right-add-neg-one-ℤ : (x : )  x +ℤ neg-one-ℤ  pred-ℤ x
  right-add-neg-one-ℤ x = inv (is-right-add-neg-one-pred-ℤ x)

Addition is associative

  associative-add-ℤ :
    (x y z : )  ((x +ℤ y) +ℤ z)  (x +ℤ (y +ℤ z))
  associative-add-ℤ (inl zero-ℕ) y z =
      (neg-one-ℤ +ℤ y) +ℤ z
       (pred-ℤ (zero-ℤ +ℤ y)) +ℤ z
        by ap (_+ℤ z) (left-predecessor-law-add-ℤ zero-ℤ y)
       pred-ℤ (y +ℤ z)
        by left-predecessor-law-add-ℤ y z
       neg-one-ℤ +ℤ (y +ℤ z)
        by inv (left-predecessor-law-add-ℤ zero-ℤ (y +ℤ z))
  associative-add-ℤ (inl (succ-ℕ x)) y z =
      (pred-ℤ (inl x) +ℤ y) +ℤ z
       pred-ℤ (inl x +ℤ y) +ℤ z
        by ap (_+ℤ z) (left-predecessor-law-add-ℤ (inl x) y)
       pred-ℤ ((inl x +ℤ y) +ℤ z)
        by left-predecessor-law-add-ℤ ((inl x) +ℤ y) z
       pred-ℤ (inl x +ℤ (y +ℤ z))
        by ap pred-ℤ (associative-add-ℤ (inl x) y z)
       pred-ℤ (inl x) +ℤ (y +ℤ z)
        by inv (left-predecessor-law-add-ℤ (inl x) (y +ℤ z))
  associative-add-ℤ (inr (inl star)) y z =
  associative-add-ℤ (inr (inr zero-ℕ)) y z =
      (one-ℤ +ℤ y) +ℤ z
       succ-ℤ (zero-ℤ +ℤ y) +ℤ z
        by ap (_+ℤ z) (left-successor-law-add-ℤ zero-ℤ y)
       succ-ℤ (y +ℤ z)
        by left-successor-law-add-ℤ y z
       one-ℤ +ℤ (y +ℤ z)
        by inv (left-successor-law-add-ℤ zero-ℤ (y +ℤ z))
  associative-add-ℤ (inr (inr (succ-ℕ x))) y z =
      (succ-ℤ (inr (inr x)) +ℤ y) +ℤ z
       succ-ℤ (inr (inr x) +ℤ y) +ℤ z
        by ap (_+ℤ z) (left-successor-law-add-ℤ (inr (inr x)) y)
       succ-ℤ ((inr (inr x) +ℤ y) +ℤ z)
        by left-successor-law-add-ℤ ((inr (inr x)) +ℤ y) z
       succ-ℤ (inr (inr x) +ℤ (y +ℤ z))
        by ap succ-ℤ (associative-add-ℤ (inr (inr x)) y z)
       succ-ℤ (inr (inr x)) +ℤ (y +ℤ z)
        by inv (left-successor-law-add-ℤ (inr (inr x)) (y +ℤ z))

Addition is commutative

  commutative-add-ℤ : (x y : )  x +ℤ y  y +ℤ x
  commutative-add-ℤ (inl zero-ℕ) y =
      neg-one-ℤ +ℤ y
       pred-ℤ (zero-ℤ +ℤ y)
        by left-predecessor-law-add-ℤ zero-ℤ y
       pred-ℤ (y +ℤ zero-ℤ)
        by inv (ap pred-ℤ (right-unit-law-add-ℤ y))
       y +ℤ neg-one-ℤ
        by inv (right-predecessor-law-add-ℤ y zero-ℤ)
  commutative-add-ℤ (inl (succ-ℕ x)) y =
      (inl (succ-ℕ x)) +ℤ y
       pred-ℤ (y +ℤ (inl x))
        by ap pred-ℤ (commutative-add-ℤ (inl x) y)
       y +ℤ (inl (succ-ℕ x))
        by inv (right-predecessor-law-add-ℤ y (inl x))
  commutative-add-ℤ (inr (inl star)) y =
    inv (right-unit-law-add-ℤ y)
  commutative-add-ℤ (inr (inr zero-ℕ)) y =
      succ-ℤ y
       succ-ℤ (y +ℤ zero-ℤ)
        by inv (ap succ-ℤ (right-unit-law-add-ℤ y))
       y +ℤ one-ℤ
        by inv (right-successor-law-add-ℤ y zero-ℤ)
  commutative-add-ℤ (inr (inr (succ-ℕ x))) y =
      succ-ℤ ((inr (inr x)) +ℤ y)
       succ-ℤ (y +ℤ (inr (inr x)))
        by ap succ-ℤ (commutative-add-ℤ (inr (inr x)) y)
       y +ℤ (succ-ℤ (inr (inr x)))
        by inv (right-successor-law-add-ℤ y (inr (inr x)))

The inverse laws for addition and negatives

  left-inverse-law-add-ℤ :
    (x : )  neg-ℤ x +ℤ x  zero-ℤ
  left-inverse-law-add-ℤ (inl zero-ℕ) = refl
  left-inverse-law-add-ℤ (inl (succ-ℕ x)) =
      succ-ℤ (inr (inr x) +ℤ pred-ℤ (inl x))
       succ-ℤ (pred-ℤ (inr (inr x) +ℤ inl x))
        by ap succ-ℤ (right-predecessor-law-add-ℤ (inr (inr x)) (inl x))
       inr (inr x) +ℤ inl x
        by is-section-pred-ℤ ((inr (inr x)) +ℤ (inl x))
        by left-inverse-law-add-ℤ (inl x)
  left-inverse-law-add-ℤ (inr (inl star)) = refl
  left-inverse-law-add-ℤ (inr (inr x)) =
      neg-ℤ (inr (inr x)) +ℤ inr (inr x)
       inr (inr x) +ℤ inl x
        by commutative-add-ℤ (inl x) (inr (inr x))
        by left-inverse-law-add-ℤ (inl x)

  right-inverse-law-add-ℤ :
    (x : )  x +ℤ neg-ℤ x  zero-ℤ
  right-inverse-law-add-ℤ x =
      x +ℤ neg-ℤ x
       neg-ℤ x +ℤ x
        by commutative-add-ℤ x (neg-ℤ x)
        by left-inverse-law-add-ℤ x

Interchange law for addition with respect to addition

  interchange-law-add-add-ℤ :
    (x y u v : )  (x +ℤ y) +ℤ (u +ℤ v)  (x +ℤ u) +ℤ (y +ℤ v)
  interchange-law-add-add-ℤ =

Addition by x is a binary equivalence

  is-section-left-add-neg-ℤ :
    (x y : )  x +ℤ (neg-ℤ x +ℤ y)  y
  is-section-left-add-neg-ℤ x y =
      x +ℤ (neg-ℤ x +ℤ y)
       (x +ℤ neg-ℤ x) +ℤ y
        by inv (associative-add-ℤ x (neg-ℤ x) y)
        by ap (_+ℤ y) (right-inverse-law-add-ℤ x)

  is-retraction-left-add-neg-ℤ :
    (x y : )  (neg-ℤ x) +ℤ (x +ℤ y)  y
  is-retraction-left-add-neg-ℤ x y =
      neg-ℤ x +ℤ (x +ℤ y)
       (neg-ℤ x +ℤ x) +ℤ y
        by inv (associative-add-ℤ (neg-ℤ x) x y)
        by ap (_+ℤ y) (left-inverse-law-add-ℤ x)

  is-equiv-left-add-ℤ : (x : )  is-equiv (x +ℤ_)
  pr1 (pr1 (is-equiv-left-add-ℤ x)) = add-ℤ (neg-ℤ x)
  pr2 (pr1 (is-equiv-left-add-ℤ x)) = is-section-left-add-neg-ℤ x
  pr1 (pr2 (is-equiv-left-add-ℤ x)) = add-ℤ (neg-ℤ x)
  pr2 (pr2 (is-equiv-left-add-ℤ x)) = is-retraction-left-add-neg-ℤ x

equiv-left-add-ℤ :   (  )
pr1 (equiv-left-add-ℤ x) = add-ℤ x
pr2 (equiv-left-add-ℤ x) = is-equiv-left-add-ℤ x

  is-section-right-add-neg-ℤ :
    (x y : )  (y +ℤ neg-ℤ x) +ℤ x  y
  is-section-right-add-neg-ℤ x y =
      (y +ℤ neg-ℤ x) +ℤ x
       y +ℤ (neg-ℤ x +ℤ x)
        by associative-add-ℤ y (neg-ℤ x) x
       y +ℤ zero-ℤ
        by ap (y +ℤ_) (left-inverse-law-add-ℤ x)
        by right-unit-law-add-ℤ y

  is-retraction-right-add-neg-ℤ :
    (x y : )  (y +ℤ x) +ℤ neg-ℤ x  y
  is-retraction-right-add-neg-ℤ x y =
      (y +ℤ x) +ℤ neg-ℤ x
       y +ℤ (x +ℤ neg-ℤ x)
        by associative-add-ℤ y x (neg-ℤ x)
       y +ℤ zero-ℤ
        by ap (y +ℤ_) (right-inverse-law-add-ℤ x)
        by right-unit-law-add-ℤ y

  is-equiv-right-add-ℤ : (y : )  is-equiv (_+ℤ y)
  pr1 (pr1 (is-equiv-right-add-ℤ y)) = _+ℤ (neg-ℤ y)
  pr2 (pr1 (is-equiv-right-add-ℤ y)) = is-section-right-add-neg-ℤ y
  pr1 (pr2 (is-equiv-right-add-ℤ y)) = _+ℤ (neg-ℤ y)
  pr2 (pr2 (is-equiv-right-add-ℤ y)) = is-retraction-right-add-neg-ℤ y

equiv-right-add-ℤ :   (  )
pr1 (equiv-right-add-ℤ y) = _+ℤ y
pr2 (equiv-right-add-ℤ y) = is-equiv-right-add-ℤ y

is-binary-equiv-left-add-ℤ : is-binary-equiv add-ℤ
pr1 is-binary-equiv-left-add-ℤ = is-equiv-right-add-ℤ
pr2 is-binary-equiv-left-add-ℤ = is-equiv-left-add-ℤ

Addition by an integer is a binary embedding

  is-emb-left-add-ℤ :
    (x : )  is-emb (x +ℤ_)
  is-emb-left-add-ℤ x =
    is-emb-is-equiv (is-equiv-left-add-ℤ x)

  is-emb-right-add-ℤ :
    (y : )  is-emb (_+ℤ y)
  is-emb-right-add-ℤ y = is-emb-is-equiv (is-equiv-right-add-ℤ y)

  is-binary-emb-add-ℤ : is-binary-emb add-ℤ
  is-binary-emb-add-ℤ =
    is-binary-emb-is-binary-equiv is-binary-equiv-left-add-ℤ

Addition by x is injective

  is-injective-right-add-ℤ : (x : )  is-injective (_+ℤ x)
  is-injective-right-add-ℤ x = is-injective-is-emb (is-emb-right-add-ℤ x)

  is-injective-left-add-ℤ : (x : )  is-injective (x +ℤ_)
  is-injective-left-add-ℤ x = is-injective-is-emb (is-emb-left-add-ℤ x)

Negative laws for addition

  right-negative-law-add-ℤ :
    (k l : )  k +ℤ neg-ℤ l  neg-ℤ (neg-ℤ k +ℤ l)
  right-negative-law-add-ℤ (inl zero-ℕ) l =
      neg-one-ℤ +ℤ neg-ℤ l
       pred-ℤ (zero-ℤ +ℤ neg-ℤ l)
        by left-predecessor-law-add-ℤ zero-ℤ (neg-ℤ l)
       neg-ℤ (succ-ℤ l)
        by pred-neg-ℤ l
  right-negative-law-add-ℤ (inl (succ-ℕ x)) l =
      pred-ℤ (inl x) +ℤ neg-ℤ l
       pred-ℤ (inl x +ℤ neg-ℤ l)
        by left-predecessor-law-add-ℤ (inl x) (neg-ℤ l)
       pred-ℤ (neg-ℤ (neg-ℤ (inl x) +ℤ l))
        by ap pred-ℤ (right-negative-law-add-ℤ (inl x) l)
       neg-ℤ (succ-ℤ (inr (inr x) +ℤ l))
        by pred-neg-ℤ (inr (inr x) +ℤ l)
  right-negative-law-add-ℤ (inr (inl star)) l =
  right-negative-law-add-ℤ (inr (inr zero-ℕ)) l =
    inv (neg-pred-ℤ l)
  right-negative-law-add-ℤ (inr (inr (succ-ℕ n))) l =
      succ-ℤ (in-pos-ℤ n) +ℤ neg-ℤ l
       succ-ℤ (in-pos-ℤ n +ℤ neg-ℤ l)
        by left-successor-law-add-ℤ (in-pos-ℤ n) (neg-ℤ l)
       succ-ℤ (neg-ℤ (neg-ℤ (inr (inr n)) +ℤ l))
        by ap succ-ℤ (right-negative-law-add-ℤ (inr (inr n)) l)
       neg-ℤ (pred-ℤ ((inl n) +ℤ l))
        by inv (neg-pred-ℤ ((inl n) +ℤ l))

Distributivity of negatives over addition

  distributive-neg-add-ℤ :
    (k l : )  neg-ℤ (k +ℤ l)  neg-ℤ k +ℤ neg-ℤ l
  distributive-neg-add-ℤ (inl zero-ℕ) l =
      neg-ℤ (inl zero-ℕ +ℤ l)
       neg-ℤ (pred-ℤ (zero-ℤ +ℤ l))
        by ap neg-ℤ (left-predecessor-law-add-ℤ zero-ℤ l)
       neg-ℤ (inl zero-ℕ) +ℤ neg-ℤ l
        by neg-pred-ℤ l
  distributive-neg-add-ℤ (inl (succ-ℕ n)) l =
      neg-ℤ (pred-ℤ (inl n +ℤ l))
       succ-ℤ (neg-ℤ (inl n +ℤ l))
        by neg-pred-ℤ (inl n +ℤ l)
       succ-ℤ (neg-ℤ (inl n) +ℤ neg-ℤ l)
        by ap succ-ℤ (distributive-neg-add-ℤ (inl n) l)
       neg-ℤ (pred-ℤ (inl n)) +ℤ neg-ℤ l
        by ap (_+ℤ (neg-ℤ l)) (inv (neg-pred-ℤ (inl n)))
  distributive-neg-add-ℤ (inr (inl star)) l =
  distributive-neg-add-ℤ (inr (inr zero-ℕ)) l =
    inv (pred-neg-ℤ l)
  distributive-neg-add-ℤ (inr (inr (succ-ℕ n))) l =
      neg-ℤ (succ-ℤ (in-pos-ℤ n +ℤ l))
       pred-ℤ (neg-ℤ (in-pos-ℤ n +ℤ l))
        by inv (pred-neg-ℤ (in-pos-ℤ n +ℤ l))
       pred-ℤ (neg-ℤ (inr (inr n)) +ℤ neg-ℤ l)
        by ap pred-ℤ (distributive-neg-add-ℤ (inr (inr n)) l)

The inclusion of ℕ into ℤ preserves addition

  add-int-ℕ : (x y : )  (int-ℕ x) +ℤ (int-ℕ y)  int-ℕ (x +ℕ y)
  add-int-ℕ x zero-ℕ = right-unit-law-add-ℤ (int-ℕ x)
  add-int-ℕ x (succ-ℕ y) =
      int-ℕ x +ℤ int-ℕ (succ-ℕ y)
       int-ℕ x +ℤ succ-ℤ (int-ℕ y)
        by ap ((int-ℕ x) +ℤ_) (inv (succ-int-ℕ y))
       succ-ℤ (int-ℕ x +ℤ int-ℕ y)
        by right-successor-law-add-ℤ (int-ℕ x) (int-ℕ y)
       succ-ℤ (int-ℕ (x +ℕ y))
        by ap succ-ℤ (add-int-ℕ x y)
       int-ℕ (succ-ℕ (x +ℕ y))
        by succ-int-ℕ (x +ℕ y)

If x + y = y then x = 0

  is-zero-left-add-ℤ :
    (x y : )  x +ℤ y  y  is-zero-ℤ x
  is-zero-left-add-ℤ x y H =
       x +ℤ zero-ℤ
        by inv (right-unit-law-add-ℤ x)
       x +ℤ (y +ℤ neg-ℤ y)
        by inv (ap (x +ℤ_) (right-inverse-law-add-ℤ y))
       (x +ℤ y) +ℤ neg-ℤ y
        by inv (associative-add-ℤ x y (neg-ℤ y))
       y +ℤ neg-ℤ y
        by ap (_+ℤ (neg-ℤ y)) H
        by right-inverse-law-add-ℤ y

If x + y = x then y = 0

  is-zero-right-add-ℤ :
    (x y : )  x +ℤ y  x  is-zero-ℤ y
  is-zero-right-add-ℤ x y H =
    is-zero-left-add-ℤ y x (commutative-add-ℤ y x  H)

See also

Recent changes