Integer fractions

Content created by Fredrik Bakke, Egbert Rijke, malarbol, Fernando Chu, Julian KG, fernabnor and louismntnu.

Created on 2023-04-08.
Last modified on 2024-04-09.

module elementary-number-theory.integer-fractions where
open import elementary-number-theory.greatest-common-divisor-integers
open import elementary-number-theory.integers
open import elementary-number-theory.multiplication-integers
open import elementary-number-theory.nonzero-integers
open import elementary-number-theory.positive-and-negative-integers
open import elementary-number-theory.positive-integers

open import foundation.action-on-identifications-functions
open import foundation.binary-relations
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.equivalence-relations
open import foundation.identity-types
open import foundation.negation
open import foundation.propositions
open import foundation.sets
open import foundation.universe-levels


The type of integer fractions is the type of pairs n/m consisting of an integer n and a positive integer m. The type of rational numbers is a retract of the type of fractions.


The type of fractions

fraction-ℤ : UU lzero
fraction-ℤ =  × positive-ℤ

The numerator of a fraction

numerator-fraction-ℤ : fraction-ℤ  
numerator-fraction-ℤ x = pr1 x

The denominator of a fraction

positive-denominator-fraction-ℤ : fraction-ℤ  positive-ℤ
positive-denominator-fraction-ℤ x = pr2 x

denominator-fraction-ℤ : fraction-ℤ  
denominator-fraction-ℤ x = pr1 (positive-denominator-fraction-ℤ x)

is-positive-denominator-fraction-ℤ :
  (x : fraction-ℤ)  is-positive-ℤ (denominator-fraction-ℤ x)
is-positive-denominator-fraction-ℤ x = pr2 (positive-denominator-fraction-ℤ x)

Inclusion of the integers

in-fraction-ℤ :   fraction-ℤ
pr1 (in-fraction-ℤ x) = x
pr2 (in-fraction-ℤ x) = one-positive-ℤ

Negative one, zero and one

neg-one-fraction-ℤ : fraction-ℤ
neg-one-fraction-ℤ = in-fraction-ℤ neg-one-ℤ

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

zero-fraction-ℤ : fraction-ℤ
zero-fraction-ℤ = in-fraction-ℤ zero-ℤ

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

is-nonzero-fraction-ℤ : fraction-ℤ  UU lzero
is-nonzero-fraction-ℤ k = ¬ (is-zero-fraction-ℤ k)

one-fraction-ℤ : fraction-ℤ
one-fraction-ℤ = in-fraction-ℤ one-ℤ

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

The negative of an integer fraction

neg-fraction-ℤ : fraction-ℤ  fraction-ℤ
neg-fraction-ℤ (d , n) = (neg-ℤ d , n)


Denominators are nonzero

  is-nonzero-denominator-fraction-ℤ :
    (x : fraction-ℤ)  is-nonzero-ℤ (denominator-fraction-ℤ x)
  is-nonzero-denominator-fraction-ℤ x =
    is-nonzero-is-positive-ℤ (is-positive-denominator-fraction-ℤ x)

The type of fractions is a set

  is-set-fraction-ℤ : is-set fraction-ℤ
  is-set-fraction-ℤ = is-set-Σ is-set-ℤ  _  is-set-positive-ℤ)
sim-fraction-ℤ-Prop : fraction-ℤ  fraction-ℤ  Prop lzero
sim-fraction-ℤ-Prop x y =
  Id-Prop ℤ-Set
    ((numerator-fraction-ℤ x) *ℤ (denominator-fraction-ℤ y))
    ((numerator-fraction-ℤ y) *ℤ (denominator-fraction-ℤ x))

sim-fraction-ℤ : fraction-ℤ  fraction-ℤ  UU lzero
sim-fraction-ℤ x y = type-Prop (sim-fraction-ℤ-Prop x y)

is-prop-sim-fraction-ℤ : (x y : fraction-ℤ)  is-prop (sim-fraction-ℤ x y)
is-prop-sim-fraction-ℤ x y = is-prop-type-Prop (sim-fraction-ℤ-Prop x y)

refl-sim-fraction-ℤ : is-reflexive sim-fraction-ℤ
refl-sim-fraction-ℤ x = refl

symmetric-sim-fraction-ℤ : is-symmetric sim-fraction-ℤ
symmetric-sim-fraction-ℤ x y r = inv r

  transitive-sim-fraction-ℤ : is-transitive sim-fraction-ℤ
  transitive-sim-fraction-ℤ x y z s r =
      ( denominator-fraction-ℤ y)
      ( is-nonzero-denominator-fraction-ℤ y)
      ( ( associative-mul-ℤ
          ( numerator-fraction-ℤ x)
          ( denominator-fraction-ℤ z)
          ( denominator-fraction-ℤ y)) 
        ( ap
          ( (numerator-fraction-ℤ x) *ℤ_)
          ( commutative-mul-ℤ
            ( denominator-fraction-ℤ z)
            ( denominator-fraction-ℤ y))) 
        ( inv
          ( associative-mul-ℤ
            ( numerator-fraction-ℤ x)
            ( denominator-fraction-ℤ y)
            ( denominator-fraction-ℤ z))) 
        ( ap ( _*ℤ (denominator-fraction-ℤ z)) r) 
        ( associative-mul-ℤ
          ( numerator-fraction-ℤ y)
          ( denominator-fraction-ℤ x)
          ( denominator-fraction-ℤ z)) 
        ( ap
          ( (numerator-fraction-ℤ y) *ℤ_)
          ( commutative-mul-ℤ
            ( denominator-fraction-ℤ x)
            ( denominator-fraction-ℤ z))) 
        ( inv
          ( associative-mul-ℤ
            ( numerator-fraction-ℤ y)
            ( denominator-fraction-ℤ z)
            ( denominator-fraction-ℤ x))) 
        ( ap (_*ℤ (denominator-fraction-ℤ x)) s) 
        ( associative-mul-ℤ
          ( numerator-fraction-ℤ z)
          ( denominator-fraction-ℤ y)
          ( denominator-fraction-ℤ x)) 
        ( ap
          ( (numerator-fraction-ℤ z) *ℤ_)
          ( commutative-mul-ℤ
            ( denominator-fraction-ℤ y)
            ( denominator-fraction-ℤ x))) 
        ( inv
          ( associative-mul-ℤ
            ( numerator-fraction-ℤ z)
            ( denominator-fraction-ℤ x)
            ( denominator-fraction-ℤ y))))

equivalence-relation-sim-fraction-ℤ : equivalence-relation lzero fraction-ℤ
pr1 equivalence-relation-sim-fraction-ℤ = sim-fraction-ℤ-Prop
pr1 (pr2 equivalence-relation-sim-fraction-ℤ) = refl-sim-fraction-ℤ
pr1 (pr2 (pr2 equivalence-relation-sim-fraction-ℤ)) = symmetric-sim-fraction-ℤ
pr2 (pr2 (pr2 equivalence-relation-sim-fraction-ℤ)) = transitive-sim-fraction-ℤ

The negatives of two similar integer fractions are similar

module _
  (x y : fraction-ℤ)

    preserves-sim-neg-fraction-ℤ :
      sim-fraction-ℤ x y  sim-fraction-ℤ (neg-fraction-ℤ x) (neg-fraction-ℤ y)
    preserves-sim-neg-fraction-ℤ H =
      ( left-negative-law-mul-ℤ
        ( numerator-fraction-ℤ x)
        ( denominator-fraction-ℤ y)) 
      ( ap neg-ℤ H) 
      ( inv
        ( left-negative-law-mul-ℤ
          ( numerator-fraction-ℤ y)
          ( denominator-fraction-ℤ x)))

Two integer fractions with zero numerator are similar

  sim-is-zero-numerator-fraction-ℤ :
    (x y : fraction-ℤ) 
    is-zero-ℤ (numerator-fraction-ℤ x) 
    is-zero-ℤ (numerator-fraction-ℤ y) 
    sim-fraction-ℤ x y
  sim-is-zero-numerator-fraction-ℤ x y H K =
    ( ap (_*ℤ (denominator-fraction-ℤ y)) H 
      left-zero-law-mul-ℤ (denominator-fraction-ℤ y))
    ( ap (_*ℤ (denominator-fraction-ℤ x)) K 
      left-zero-law-mul-ℤ (denominator-fraction-ℤ x))

The greatest common divisor of the numerator and a denominator of a fraction is always a positive integer

  is-positive-gcd-numerator-denominator-fraction-ℤ :
    (x : fraction-ℤ) 
    is-positive-ℤ (gcd-ℤ (numerator-fraction-ℤ x) (denominator-fraction-ℤ x))
  is-positive-gcd-numerator-denominator-fraction-ℤ x =
      ( numerator-fraction-ℤ x)
      ( denominator-fraction-ℤ x)
      ( is-positive-denominator-fraction-ℤ x)

  is-nonzero-gcd-numerator-denominator-fraction-ℤ :
    (x : fraction-ℤ) 
    is-nonzero-ℤ (gcd-ℤ (numerator-fraction-ℤ x) (denominator-fraction-ℤ x))
  is-nonzero-gcd-numerator-denominator-fraction-ℤ x =
      ( is-positive-gcd-numerator-denominator-fraction-ℤ x)

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