Inequality on integer fractions
Content created by Fredrik Bakke, malarbol, Egbert Rijke and Fernando Chu.
Created on 2023-04-08.
Last modified on 2024-04-09.
module elementary-number-theory.inequality-integer-fractions where
Imports
open import elementary-number-theory.addition-integers open import elementary-number-theory.addition-positive-and-negative-integers open import elementary-number-theory.cross-multiplication-difference-integer-fractions open import elementary-number-theory.difference-integers open import elementary-number-theory.inequality-integers open import elementary-number-theory.integer-fractions open import elementary-number-theory.integers open import elementary-number-theory.multiplication-integers open import elementary-number-theory.multiplication-positive-and-negative-integers open import elementary-number-theory.nonnegative-integers open import elementary-number-theory.nonpositive-integers open import elementary-number-theory.positive-and-negative-integers open import elementary-number-theory.positive-integers open import elementary-number-theory.strict-inequality-integers open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.dependent-pair-types open import foundation.function-types open import foundation.identity-types open import foundation.negation open import foundation.propositions open import foundation.transport-along-identifications open import foundation.universe-levels
Idea
An integer fraction m/n
is
less or equal¶
to a fraction m'/n'
if the
integer product m * n'
is less or equal to m' * n
.
Definition
Inequality on integer fractions
leq-fraction-ℤ-Prop : fraction-ℤ → fraction-ℤ → Prop lzero leq-fraction-ℤ-Prop (m , n , p) (m' , n' , p') = leq-ℤ-Prop (m *ℤ n') (m' *ℤ n) leq-fraction-ℤ : fraction-ℤ → fraction-ℤ → UU lzero leq-fraction-ℤ x y = type-Prop (leq-fraction-ℤ-Prop x y) is-prop-leq-fraction-ℤ : (x y : fraction-ℤ) → is-prop (leq-fraction-ℤ x y) is-prop-leq-fraction-ℤ x y = is-prop-type-Prop (leq-fraction-ℤ-Prop x y)
Properties
Inequality on integer fractions is decidable
is-decidable-leq-fraction-ℤ : (x y : fraction-ℤ) → (leq-fraction-ℤ x y) + ¬ (leq-fraction-ℤ x y) is-decidable-leq-fraction-ℤ x y = is-decidable-leq-ℤ ( numerator-fraction-ℤ x *ℤ denominator-fraction-ℤ y) ( numerator-fraction-ℤ y *ℤ denominator-fraction-ℤ x) leq-fraction-ℤ-Decidable-Prop : (x y : fraction-ℤ) → Decidable-Prop lzero leq-fraction-ℤ-Decidable-Prop x y = ( leq-fraction-ℤ x y , is-prop-leq-fraction-ℤ x y , is-decidable-leq-fraction-ℤ x y)
Inequality on integer fractions is antisymmetric with respect to the similarity relation
module _ (x y : fraction-ℤ) where is-sim-antisymmetric-leq-fraction-ℤ : leq-fraction-ℤ x y → leq-fraction-ℤ y x → sim-fraction-ℤ x y is-sim-antisymmetric-leq-fraction-ℤ H H' = sim-is-zero-coss-mul-diff-fraction-ℤ x y ( is-zero-is-nonnegative-is-nonpositive-ℤ ( H) ( is-nonpositive-eq-ℤ ( skew-commutative-cross-mul-diff-fraction-ℤ y x) ( is-nonpositive-neg-is-nonnegative-ℤ ( H'))))
Inequality on integer fractions is transitive
transitive-leq-fraction-ℤ : (p q r : fraction-ℤ) → leq-fraction-ℤ q r → leq-fraction-ℤ p q → leq-fraction-ℤ p r transitive-leq-fraction-ℤ p q r H H' = is-nonnegative-right-factor-mul-ℤ ( is-nonnegative-eq-ℤ ( lemma-add-cross-mul-diff-fraction-ℤ p q r) ( is-nonnegative-add-ℤ ( is-nonnegative-mul-ℤ ( is-nonnegative-is-positive-ℤ ( is-positive-denominator-fraction-ℤ p)) ( H)) ( is-nonnegative-mul-ℤ ( is-nonnegative-is-positive-ℤ ( is-positive-denominator-fraction-ℤ r)) ( H')))) ( is-positive-denominator-fraction-ℤ q)
Recent changes
- 2024-04-09. malarbol and Fredrik Bakke. The additive group of rational numbers (#1100).
- 2024-03-28. malarbol and Fredrik Bakke. Refactoring positive integers (#1059).
- 2024-02-27. malarbol. Rational real numbers (#1034).
- 2023-05-13. Fredrik Bakke. Refactor to use infix binary operators for arithmetic (#620).
- 2023-04-08. Egbert Rijke. Refactoring elementary number theory files (#546).