Skip to content

Inserting and removing elements in finite sequences#1929

Open
malarbol wants to merge 33 commits intoUniMath:masterfrom
malarbol:pi-fin-seq
Open

Inserting and removing elements in finite sequences#1929
malarbol wants to merge 33 commits intoUniMath:masterfrom
malarbol:pi-fin-seq

Conversation

@malarbol
Copy link
Copy Markdown
Collaborator

This PR introduces the following maps for finite sequences:

  • (x₀,...xᵢ₋₁,xᵢ,xᵢ₊₁,...,xₙ) ↦ xᵢ;
  • (x₀,...xᵢ₋₁,xᵢ,xᵢ₊₁,...,xₙ) ↦ (x₀,...xᵢ₋₁,xᵢ₊₁,...,xₙ);
  • xᵢ ↦ (x₀,...xᵢ₋₁,xᵢ₊₁,...,xₙ) ↦ (x₀,...xᵢ₋₁,xᵢ,xᵢ₊₁,...,xₙ);

and the corresponding equivalences Aⁿ⁺¹ ≃ A × Aⁿ:

  • (x₀,...xᵢ₋₁,xᵢ,xᵢ₊₁,...,xₙ) ↔ (xᵢ , (x₀,...xᵢ₋₁,xᵢ₊₁,...,xₙ)).

We apply these to the case of finite sequences of types A : Fin n → UU l and define

  Πₙ A = (i : Fin n) → A i ≃ A₀ × A₁ × ... × Aᵢ × ... Aₙ₋₁

to prove that

  Πₙ₊₁ A ≃ Aᵢ × Πₙ Aⁱ

where Aⁱ denotes the finite sequence of types obtained by removing the ith component of A so

Πₙ Aⁱ = A₀ × ... Aᵢ₋₁ × Aᵢ₊ᵢ × ... × Aₙ.

@malarbol malarbol added the lists label Mar 30, 2026
Comment thread src/lists/insert-at-index-finite-sequences.lagda.md Outdated
Sign up for free to join this conversation on GitHub. Already have an account? Sign in to comment

Labels

Projects

None yet

Development

Successfully merging this pull request may close these issues.

2 participants