Equality in the standard finite types

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

Created on 2022-02-28.
Last modified on 2023-06-10.

module univalent-combinatorics.equality-standard-finite-types where
open import elementary-number-theory.inequality-natural-numbers
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.apartness-relations
open import foundation.contractible-types
open import foundation.coproduct-types
open import foundation.decidable-types
open import foundation.dependent-pair-types
open import foundation.discrete-types
open import foundation.empty-types
open import foundation.equivalences
open import foundation.functoriality-coproduct-types
open import foundation.identity-types
open import foundation.negation
open import foundation.propositions
open import foundation.set-truncations
open import foundation.tight-apartness-relations
open import foundation.unit-type
open import foundation.universe-levels

open import foundation-core.decidable-propositions

open import univalent-combinatorics.standard-finite-types


Since the standard finite types are defined recursively by adding a point one at a time, it follows that equality in the standard finite types is decidable, and that they are sets.


Characterization of the identity types of the standard finite types

Eq-Fin : (k : )  Fin k  Fin k  UU lzero
Eq-Fin (succ-ℕ k) (inl x) (inl y) = Eq-Fin k x y
Eq-Fin (succ-ℕ k) (inl x) (inr y) = empty
Eq-Fin (succ-ℕ k) (inr x) (inl y) = empty
Eq-Fin (succ-ℕ k) (inr x) (inr y) = unit

is-prop-Eq-Fin : (k : )  (x : Fin k)  (y : Fin k)  is-prop (Eq-Fin k x y)
is-prop-Eq-Fin (succ-ℕ k) (inl x) (inl y) = is-prop-Eq-Fin k x y
is-prop-Eq-Fin (succ-ℕ k) (inr x) (inl y) = is-prop-empty
is-prop-Eq-Fin (succ-ℕ k) (inl x) (inr y) = is-prop-empty
is-prop-Eq-Fin (succ-ℕ k) (inr x) (inr y) = is-prop-unit

refl-Eq-Fin : (k : ) (x : Fin k)  Eq-Fin k x x
refl-Eq-Fin (succ-ℕ k) (inl x) = refl-Eq-Fin k x
refl-Eq-Fin (succ-ℕ k) (inr x) = star

Eq-Fin-eq : (k : ) {x y : Fin k}  Id x y  Eq-Fin k x y
Eq-Fin-eq k refl = refl-Eq-Fin k _

eq-Eq-Fin :
  (k : ) {x y : Fin k}  Eq-Fin k x y  Id x y
eq-Eq-Fin (succ-ℕ k) {inl x} {inl y} e = ap inl (eq-Eq-Fin k e)
eq-Eq-Fin (succ-ℕ k) {inr star} {inr star} star = refl

extensionality-Fin :
  (k : )
  (x y : Fin k) 
  (x  y)  (Eq-Fin k x y)
pr1 (extensionality-Fin k x y) = Eq-Fin-eq k
pr2 (extensionality-Fin k x y) =
    ( is-set-Fin k x y)
    ( is-prop-Eq-Fin k x y)
    ( eq-Eq-Fin k)

is-decidable-Eq-Fin : (k : ) (x y : Fin k)  is-decidable (Eq-Fin k x y)
is-decidable-Eq-Fin (succ-ℕ k) (inl x) (inl y) = is-decidable-Eq-Fin k x y
is-decidable-Eq-Fin (succ-ℕ k) (inl x) (inr y) = is-decidable-empty
is-decidable-Eq-Fin (succ-ℕ k) (inr x) (inl y) = is-decidable-empty
is-decidable-Eq-Fin (succ-ℕ k) (inr x) (inr y) = is-decidable-unit

has-decidable-equality-Fin :
  (k : ) (x y : Fin k)  is-decidable (Id x y)
has-decidable-equality-Fin k x y =
  map-coprod (eq-Eq-Fin k) (map-neg (Eq-Fin-eq k)) (is-decidable-Eq-Fin k x y)

Fin-Discrete-Type :   Discrete-Type lzero
pr1 (Fin-Discrete-Type k) = Fin k
pr2 (Fin-Discrete-Type k) = has-decidable-equality-Fin k

is-decidable-is-zero-Fin :
  {k : } (x : Fin k)  is-decidable (is-zero-Fin k x)
is-decidable-is-zero-Fin {succ-ℕ k} x =
  has-decidable-equality-Fin (succ-ℕ k) x (zero-Fin k)

is-decidable-is-neg-one-Fin :
  {k : } (x : Fin k)  is-decidable (is-neg-one-Fin k x)
is-decidable-is-neg-one-Fin {succ-ℕ k} x =
  has-decidable-equality-Fin (succ-ℕ k) x (neg-one-Fin k)

is-decidable-is-one-Fin :
  {k : } (x : Fin k)  is-decidable (is-one-Fin k x)
is-decidable-is-one-Fin {succ-ℕ k} x =
  has-decidable-equality-Fin (succ-ℕ k) x (one-Fin k)

Being zero or being one is a proposition

is-prop-is-zero-Fin :
  (k : ) (x : Fin (succ-ℕ k))  is-prop (is-zero-Fin (succ-ℕ k) x)
is-prop-is-zero-Fin k x = is-set-Fin (succ-ℕ k) x (zero-Fin k)

is-prop-is-one-Fin :
  (k : ) (x : Fin (succ-ℕ k))  is-prop (is-one-Fin (succ-ℕ k) x)
is-prop-is-one-Fin k x = is-set-Fin (succ-ℕ k) x (one-Fin k)

is-prop-is-zero-or-one-Fin-two-ℕ :
  (x : Fin 2)  is-prop ((is-zero-Fin 2 x) + (is-one-Fin 2 x))
is-prop-is-zero-or-one-Fin-two-ℕ x =
    ( λ p q  Eq-Fin-eq 2 (inv p  q))
    ( is-prop-is-zero-Fin 1 x)
    ( is-prop-is-one-Fin 1 x)

Every element in the standard two-element type is either 0 or 1

is-contr-is-zero-or-one-Fin-two-ℕ :
  (x : Fin 2)  is-contr ((is-zero-Fin 2 x) + (is-one-Fin 2 x))
is-contr-is-zero-or-one-Fin-two-ℕ x =
    ( is-prop-is-zero-or-one-Fin-two-ℕ x)
    ( is-zero-or-one-Fin-two-ℕ x)
decidable-Eq-Fin :
  (n : ) (i j : Fin n)  Decidable-Prop lzero
pr1 (decidable-Eq-Fin n i j) = Id i j
pr1 (pr2 (decidable-Eq-Fin n i j)) = is-set-Fin n i j
pr2 (pr2 (decidable-Eq-Fin n i j)) = has-decidable-equality-Fin n i j

The standard finite types are their own set truncations

equiv-unit-trunc-Fin-Set : (k : )  Fin k  type-trunc-Set (Fin k)
equiv-unit-trunc-Fin-Set k = equiv-unit-trunc-Set (Fin-Set k)

If leq-ℕ 2 n, then there exists two distinct elements in Fin n

two-distinct-elements-leq-2-Fin :
  (n : )  leq-ℕ 2 n  Σ (Fin n)  x  Σ (Fin n)  y  ¬ (Id x y)))
pr1 (two-distinct-elements-leq-2-Fin (succ-ℕ (succ-ℕ n)) ineq) =
  inr star
pr1 (pr2 (two-distinct-elements-leq-2-Fin (succ-ℕ (succ-ℕ n)) ineq)) =
  inl (inr star)
pr2 (pr2 (two-distinct-elements-leq-2-Fin (succ-ℕ (succ-ℕ n)) ineq)) =

The standard finite type with a (tight) apartness relation

Fin-Type-With-Apartness : (k : )  Type-With-Apartness lzero lzero
Fin-Type-With-Apartness k =
  type-with-apartness-Discrete-Type (Fin-Discrete-Type k)

Fin-Type-With-Tight-Apartness : (k : )  Type-With-Tight-Apartness lzero lzero
Fin-Type-With-Tight-Apartness k =
  type-with-tight-apartness-Discrete-Type (Fin-Discrete-Type k)

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