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[Merged by Bors] - doc: add docstrings for reversed range telescoping lemmas #39605

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[Merged by Bors] - doc: add docstrings for reversed range telescoping lemmas #39605

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10 changes: 8 additions & 2 deletions Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
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Expand Up @@ -896,11 +896,17 @@ additive group reduces to the difference of the last and first terms. -/]
lemma prod_range_div (f : ℕ → G) (n : ℕ) : (∏ i ∈ range n, f (i + 1) / f i) = f n / f 0 := by
apply prod_range_induction <;> simp

@[to_additive]
/-- A reversed telescoping product along `{0, ..., n - 1}` of a commutative-group-valued function
reduces to the ratio of the first and last factors. -/
@[to_additive /-- A reversed telescoping sum along `{0, ..., n - 1}` of a function valued in a
commutative additive group reduces to the difference of the first and last terms. -/]
lemma prod_range_div' (f : ℕ → G) (n : ℕ) : (∏ i ∈ range n, f i / f (i + 1)) = f 0 / f n := by
apply prod_range_induction <;> simp

@[to_additive]
/-- Express `f n` as `f 0` multiplied by the telescoping product of consecutive ratios from
`0` to `n - 1`. -/
@[to_additive /-- Express `f n` as `f 0` plus the telescoping sum of consecutive differences from
`0` to `n - 1`. -/]
lemma eq_prod_range_div (f : ℕ → G) (n : ℕ) : f n = f 0 * ∏ i ∈ range n, f (i + 1) / f i := by
rw [prod_range_div, mul_div_cancel]

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