prod_range_succ_div_prod — Mathlib

English

Rearrangement identity for double sums over Ico: ∑_{i∈Ico a b} ∑_{j∈Ico i b} f(i,j) = ∑_{j∈Ico a b} ∑_{i∈Ico a (j+1)} f(i,j).

Русский

Перестановка двойной суммы над Ico.

LaTeX

$$$$ \\displaystyle \\sum_{i \\in \\mathrm{Ico}(a,b)} \\sum_{j \\in \\mathrm{Ico}(i,b)} f(i,j) = \\sum_{j \\in \\mathrm{Ico}(a,b)} \\sum_{i \\in \\mathrm{Ico}(a,j+1)} f(i,j). $$$$

Lean4

@[to_additive]
theorem prod_range_succ_div_prod : ((∏ i ∈ range (n + 1), f i) / ∏ i ∈ range n, f i) = f n :=
  div_eq_iff_eq_mul'.mpr <| prod_range_succ f n

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