feat(AlgebraicTopology): SimplexCategory.toTop_map_δ_apply

[erdOne]

---

[github-actions]

PR summary 5289bf289a

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ _root_.stdSimplex.map_δ_apply
+ castSucc_eq_zero
+ succAbove_eq_iff
+ succAbove_eq_zero
+ toTop_map_δ_apply
+ δ_apply
+ σ_apply

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[Raph-DG]

awaiting-author

[dagurtomas]

Thanks!

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by dagurtomas.

[joelriou]

This works also fine:

Suggested change

	lemma δ_apply {n : ℕ} (i : Fin (n + 2)) (j : Fin (⦋n⦌.len + 1)) : δ i j = i.succAbove j := rfl
	lemma δ_apply {n : ℕ} (i : Fin (n + 2)) (j : Fin (n + 1)) : δ i j = i.succAbove j := rfl

[erdOne]

This is not reducibly type correct because currently ⦋n⦌.len is not reducibly defeq to n and it causes troubles especially with the new transparency changes.

[joelriou]

Suggested change

	lemma _root_.stdSimplex.map_δ_apply {n : ℕ} (i : Fin (n + 2)) (j : Fin (⦋n + 1⦌.len + 1))
	(σ : stdSimplex ℝ (Fin (⦋n⦌.len + 1))) :
	stdSimplex.map (SimplexCategory.δ i) σ j =
	(if i < j then σ ⟨j - 1, by simpa using j.2⟩ else 0) +
	(if h : i > j then σ ⟨j, by simp_all; lia⟩ else 0) := by
	lemma _root_.stdSimplex.map_δ_apply {n : ℕ} (i j : Fin (n + 2))
	(σ : stdSimplex ℝ (Fin (⦋n⦌.len + 1))) :
	stdSimplex.map (SimplexCategory.δ i) σ j =
	(if i < j then σ ⟨j - 1, by simp_all; lia⟩ else 0) +
	(if h : i > j then σ ⟨j, by simp_all; lia⟩ else 0) := by