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Fix abs_finite_series_le signature and add Exercise 7.1.6/7.1.7 state… #448

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teorth merged 1 commit into from
Feb 19, 2026
+15 −2
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Fix abs_finite_series_le signature and add Exercise 7.1.6/7.1.7 state… #448

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17 changes: 15 additions & 2 deletions analysis/Analysis/Section_7_1.lean
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Expand Up @@ -80,11 +80,11 @@ theorem finite_series_add {m n:ℤ} (a b: ℤ → ℝ) :
∑ i ∈ Icc m n, (a i + b i) = ∑ i ∈ Icc m n, a i + ∑ i ∈ Icc m n, b i := by sorry

/-- Lemma 7.1.4(d) / Exercise 7.1.1 -/
theorem finite_series_const_mul {m n:ℤ} (a: ℤ → ℝ) (c:ℝ) :
theorem finite_series_const_mul {m n:ℤ} (a: ℤ → ℝ) (c:ℝ) :
∑ i ∈ Icc m n, c * a i = c * ∑ i ∈ Icc m n, a i := by sorry

/-- Lemma 7.1.4(e) / Exercise 7.1.1 -/
theorem abs_finite_series_le {m n:ℤ} (a: ℤ → ℝ) (c:ℝ) :
theorem abs_finite_series_le {m n:ℤ} (a: ℤ → ℝ) :
|∑ i ∈ Icc m n, a i| ≤ ∑ i ∈ Icc m n, |a i| := by sorry

/-- Lemma 7.1.4(f) / Exercise 7.1.1 -/
Expand Down Expand Up @@ -307,6 +307,19 @@ theorem lim_of_finite_series {X:Type*} [Fintype X] (a: X → ℕ → ℝ) (L : X
Filter.atTop.Tendsto (fun n ↦ ∑ x, a x n) (nhds (∑ x, L x)) := by
sorry

/-- Exercise 7.1.6 -/
theorem sum_union_disjoint {n : ℕ} {S : Type*} [Fintype S]
(E : Fin n → Finset S)
(disj : ∀ i j : Fin n, i ≠ j → Disjoint (E i) (E j))
(cover : ∀ s : S, ∃ i, s ∈ E i)
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@rkirov rkirov Feb 15, 2026

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There were a few different ways to write this in type theory, e.g. using explicit call to Finset.biUnion. Let me know if another one is preferred. I thought this one is the cleanest.

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@rkirov rkirov Feb 15, 2026

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I ended up writing a whole blog post based on this https://rkirov.github.io/posts/sets-vs-types/

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@rkirov rkirov Feb 16, 2026

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the mathlib version is here https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Basic.html#Finset.sum_biUnion but it strikes me as unnecessarily general for the purposes of this book.

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@teorth teorth Feb 19, 2026

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Looks good to me.

(f : S → ℝ) :
∑ s, f s = ∑ i, ∑ s ∈ E i, f s := by
sorry

/-- Exercise 7.1.7. Uses `Fin m` (so `aᵢ < m`) instead of the book's `aᵢ ≤ m`;
the bound is baked into the type, and `<` replaces `≤` to match the 0-indexed shift. -/
theorem sum_finite_col_row_counts {n m : ℕ} (a : Fin n → Fin m) :
∑ i, (a i : ℕ) = ∑ j : Fin m, {i : Fin n | j < a i}.toFinset.card := by
sorry

end Finset