feat(Data/Nat): add prime divisibility for ascFactorial and choose

[akiezun]

Adds two prime-divisibility lemmas for natural-number factorial/binomial APIs.

The first characterizes when a prime divides an ascending factorial:
Nat.Prime.dvd_ascFactorial_iff.

The second applies this to binomial coefficients:
Nat.Prime.dvd_choose_add_sub_one_iff, using
Nat.ascFactorial_eq_factorial_mul_choose' and cancellation of the n! factor
when n < p.

---

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[github-actions]

PR summary e7eedc30e9

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ Prime.dvd_ascFactorial_iff
+ dvd_choose_add_sub_one_iff

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

---

No changes to strong technical debt.

No changes to weak technical debt.

[wwylele]

Suggested change

	lemma Prime.dvd_ascFactorial_iff {p k n : ℕ} (hp : p.Prime) :
	p ∣ k.ascFactorial n ↔ ∃ i < n, p ∣ k + i := by
	induction n with
	| zero => simp [Nat.ascFactorial_zero, hp.ne_one]
	| succ n ih =>
	rw [Nat.ascFactorial_succ, hp.dvd_mul, ih]
	constructor
	· rintro (h | ⟨i, hi, h⟩)
	· exact ⟨n, lt_succ_self n, h⟩
	· exact ⟨i, lt_succ_of_lt hi, h⟩
	· rintro ⟨i, hi, h⟩
	rcases lt_or_eq_of_le (Nat.le_of_lt_succ hi) with hi | rfl
	· exact Or.inr ⟨i, hi, h⟩
	· exact Or.inl h
	lemma Prime.dvd_ascFactorial_iff {p k n : ℕ} (hp : p.Prime) :
	p ∣ k.ascFactorial n ↔ ∃ i < n, p ∣ k + i := by
	simp [Nat.ascFactorial_eq_prod_range, hp.prime.dvd_finsetProd_iff]

Not sure if this file has enough imports for this. If not, this can be moved to anywhere that imports Mathlib.Algebra.BigOperators.Associated and Mathlib.Data.Nat.Factorial.BigOperators

[SnirBroshi]

It doesn't and AFAICT there's no file that imports these which is also a sensible place for such a theorem.
New file?

[akiezun]

It doesn't and AFAICT there's no file that imports these which is also a sensible place for such a theorem. New file?

Thanks. Suggestions for file name?

[SnirBroshi]

Maybe Data/Nat/Factorial/Prime.lean, not sure

[wwylele]

It is going to be a bit confusing if we have both Mathlib/Data/Nat/Prime/Factorial.lean and Data/Nat/Factorial/Prime.lean... what about adding more imports here and eat the cost of large imports? (assuming it is not circular)