Multiset-indexed dependent products of types

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-10-22.
Last modified on 2023-11-24.

module trees.multiset-indexed-dependent-products-of-types where
open import elementary-number-theory.natural-numbers

open import foundation.universe-levels

open import trees.multisets
open import trees.w-types


Consider a multiset M. Then M can be seen as a tower of type families, via the inclusion from the type of all multisets, which are the well-founded trees, into the type of all trees.

This leads to the idea that we should be able to take the iterated dependent product of this tower of type families.


The iterated dependent product of types indexed by a multiset

iterated-Π-𝕍 : {l : Level}    𝕍 l  UU l
iterated-Π-𝕍 zero-ℕ (tree-𝕎 X Y) = X
iterated-Π-𝕍 (succ-ℕ n) (tree-𝕎 X Y) = (x : X)  iterated-Π-𝕍 n (Y x)

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