Addition on the natural numbers

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

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

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

open import foundation.action-on-identifications-binary-functions
open import foundation.action-on-identifications-functions
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.empty-types
open import foundation.function-types
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.interchange-law
open import foundation.negated-equality
open import foundation.sets


add-ℕ :     
add-ℕ x 0 = x
add-ℕ x (succ-ℕ y) = succ-ℕ (add-ℕ x y)

infixl 35 _+ℕ_
_+ℕ_ = add-ℕ

add-ℕ' :     
add-ℕ' m n = add-ℕ n m

ap-add-ℕ :
  {m n m' n' : }  m  m'  n  n'  m +ℕ n  m' +ℕ n'
ap-add-ℕ p q = ap-binary add-ℕ p q


The left and right unit laws

right-unit-law-add-ℕ :
  (x : )  x +ℕ zero-ℕ  x
right-unit-law-add-ℕ x = refl

left-unit-law-add-ℕ :
  (x : )  zero-ℕ +ℕ x  x
left-unit-law-add-ℕ zero-ℕ = refl
left-unit-law-add-ℕ (succ-ℕ x) = ap succ-ℕ (left-unit-law-add-ℕ x)

The left and right successor laws

  left-successor-law-add-ℕ :
    (x y : )  (succ-ℕ x) +ℕ y  succ-ℕ (x +ℕ y)
  left-successor-law-add-ℕ x zero-ℕ = refl
  left-successor-law-add-ℕ x (succ-ℕ y) =
    ap succ-ℕ (left-successor-law-add-ℕ x y)

right-successor-law-add-ℕ :
  (x y : )  x +ℕ (succ-ℕ y)  succ-ℕ (x +ℕ y)
right-successor-law-add-ℕ x y = refl

Addition is associative

  associative-add-ℕ :
    (x y z : )  (x +ℕ y) +ℕ z  x +ℕ (y +ℕ z)
  associative-add-ℕ x y zero-ℕ = refl
  associative-add-ℕ x y (succ-ℕ z) = ap succ-ℕ (associative-add-ℕ x y z)

Addition is commutative

  commutative-add-ℕ : (x y : )  x +ℕ y  y +ℕ x
  commutative-add-ℕ zero-ℕ y = left-unit-law-add-ℕ y
  commutative-add-ℕ (succ-ℕ x) y =
    (left-successor-law-add-ℕ x y)  (ap succ-ℕ (commutative-add-ℕ x y))

Addition by 1 on the left or on the right is the successor

  left-one-law-add-ℕ :
    (x : )  1 +ℕ x  succ-ℕ x
  left-one-law-add-ℕ x =
    ( left-successor-law-add-ℕ zero-ℕ x) 
    ( ap succ-ℕ (left-unit-law-add-ℕ x))

  right-one-law-add-ℕ :
    (x : )  x +ℕ 1  succ-ℕ x
  right-one-law-add-ℕ x = refl

Addition by 1 on the left or on the right is the double successor

  left-two-law-add-ℕ :
    (x : )  2 +ℕ x  succ-ℕ (succ-ℕ x)
  left-two-law-add-ℕ x =
    ( left-successor-law-add-ℕ 1 x) 
    ( ap succ-ℕ (left-one-law-add-ℕ x))

  right-two-law-add-ℕ :
    (x : )  x +ℕ 2  succ-ℕ (succ-ℕ x)
  right-two-law-add-ℕ x = refl

Interchange law of addition

  interchange-law-add-add-ℕ : interchange-law add-ℕ add-ℕ
  interchange-law-add-add-ℕ =

Addition by a fixed element on either side is injective

  is-injective-right-add-ℕ : (k : )  is-injective (_+ℕ k)
  is-injective-right-add-ℕ zero-ℕ = id
  is-injective-right-add-ℕ (succ-ℕ k) p =
    is-injective-right-add-ℕ k (is-injective-succ-ℕ p)

  is-injective-left-add-ℕ : (k : )  is-injective (k +ℕ_)
  is-injective-left-add-ℕ k {x} {y} p =
      ( k)
      ( commutative-add-ℕ x k  (p  commutative-add-ℕ k y))

Addition by a fixed element on either side is an embedding

  is-emb-left-add-ℕ : (x : )  is-emb (x +ℕ_)
  is-emb-left-add-ℕ x =
    is-emb-is-injective is-set-ℕ (is-injective-left-add-ℕ x)

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

x + y = 0 if and only if both x and y are 0

  is-zero-right-is-zero-add-ℕ :
    (x y : )  is-zero-ℕ (x +ℕ y)  is-zero-ℕ y
  is-zero-right-is-zero-add-ℕ x zero-ℕ p = refl
  is-zero-right-is-zero-add-ℕ x (succ-ℕ y) p =
    ex-falso (is-nonzero-succ-ℕ (x +ℕ y) p)

  is-zero-left-is-zero-add-ℕ :
    (x y : )  is-zero-ℕ (x +ℕ y)  is-zero-ℕ x
  is-zero-left-is-zero-add-ℕ x y p =
    is-zero-right-is-zero-add-ℕ y x ((commutative-add-ℕ y x)  p)

  is-zero-summand-is-zero-sum-ℕ :
    (x y : )  is-zero-ℕ (x +ℕ y)  (is-zero-ℕ x) × (is-zero-ℕ y)
  is-zero-summand-is-zero-sum-ℕ x y p =
    pair (is-zero-left-is-zero-add-ℕ x y p) (is-zero-right-is-zero-add-ℕ x y p)

  is-zero-sum-is-zero-summand-ℕ :
    (x y : )  (is-zero-ℕ x) × (is-zero-ℕ y)  is-zero-ℕ (x +ℕ y)
  is-zero-sum-is-zero-summand-ℕ .zero-ℕ .zero-ℕ (pair refl refl) = refl

m ≠ m + n + 1

  neq-add-ℕ :
    (m n : )  m  m +ℕ (succ-ℕ n)
  neq-add-ℕ (succ-ℕ m) n p =
    neq-add-ℕ m n
      ( ( is-injective-succ-ℕ p) 
        ( left-successor-law-add-ℕ m n))

See also

Recent changes