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[Merged by Bors] - feat(Factorial): k! divides the product of any k successive integers #21332

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[Merged by Bors] - feat(Factorial): k! divides the product of any k successive integers #21332

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feat(Factorial): k! divides the product of any k successive integers
Julian Feb 1, 2025
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Apply suggestions from code review
Julian Feb 2, 2025
f4e723f
Update Mathlib/Data/Nat/Factorial/BigOperators.lean
Julian Feb 4, 2025
164ec3d
Axe the now-unneeded helper lemma.
Julian Feb 4, 2025
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Julian Feb 5, 2025
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11 changes: 11 additions & 0 deletions Mathlib/Data/Nat/Factorial/BigOperators.lean
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Expand Up @@ -5,6 +5,7 @@ Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Tactic.Zify

/-!
# Factorial with big operators
Expand Down Expand Up @@ -41,4 +42,14 @@ theorem descFactorial_eq_prod_range (n : ℕ) : ∀ k, n.descFactorial k = ∏ i
| 0 => rfl
| k + 1 => by rw [descFactorial, prod_range_succ, mul_comm, descFactorial_eq_prod_range n k]

/-- `k!` divides the product of any `k` consecutive integers. -/
lemma factorial_coe_dvd_prod (k : ℕ) (n : ℤ) : (k ! : ℤ) ∣ ∏ i ∈ range k, (n + i) := by
rw [Int.dvd_iff_emod_eq_zero, Finset.prod_int_mod]
simp_rw [← Int.emod_add_emod n]
have hn : 0 ≤ n % k ! := Int.emod_nonneg n <| Int.natCast_ne_zero.mpr k.factorial_ne_zero
obtain ⟨x, hx⟩ := Int.eq_ofNat_of_zero_le hn
have hdivk := x.factorial_dvd_ascFactorial k
zify [x.ascFactorial_eq_prod_range k] at hdivk
rwa [← Finset.prod_int_mod, ← Int.dvd_iff_emod_eq_zero, hx]

end Nat