Addition on the natural numbers

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

open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.empty-types
open import foundation.functions
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.interchange-law
open import foundation.negation

Definition

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

add-ℕ' : ℕ → ℕ → ℕ
add-ℕ' m n = add-ℕ n m

infix 30 _+ℕ_
_+ℕ_ = add-ℕ

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

Properties

The left and right unit laws

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

left-unit-law-add-ℕ :
  (x : ℕ) → 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 : ℕ) → (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 : ℕ) → 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 : ℕ) → (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 : ℕ) → 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 : ℕ) → 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 : ℕ) → 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 : ℕ) → 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 : ℕ) → 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 =
  interchange-law-commutative-and-associative
    add-ℕ
    commutative-add-ℕ
    associative-add-ℕ

Addition by a fixed element on either side is injective

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