Double counting

module univalent-combinatorics.double-counting where

open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.identity-types
open import foundation.universe-levels

open import univalent-combinatorics.counting
open import univalent-combinatorics.standard-finite-types

Idea

Given two countings of the same type, we obtain the same number of its elements. Likewise, given two countings of equivalent types, we obtain the same number of their elements.

abstract
  double-counting-equiv :
    {l1 l2 : Level} {A : UU l1} {B : UU l2} (count-A : count A)
    (count-B : count B) (e : A ≃ B) →
    Id (number-of-elements-count count-A) (number-of-elements-count count-B)
  double-counting-equiv (pair k f) (pair l g) e =
    is-injective-Fin ((inv-equiv g ∘e e) ∘e f)

abstract
  double-counting :
    {l : Level} {A : UU l} (count-A count-A' : count A) →
    Id (number-of-elements-count count-A) (number-of-elements-count count-A')
  double-counting count-A count-A' =
    double-counting-equiv count-A count-A' id-equiv