diff --git a/Analysis/Section_11_6.lean b/Analysis/Section_11_6.lean index a5572415..8e3aa396 100644 --- a/Analysis/Section_11_6.lean +++ b/Analysis/Section_11_6.lean @@ -154,10 +154,11 @@ theorem summable_iff_integ_of_antitone {f:ℝ → ℝ} (hnon: ∀ x ≥ 0, f x -- Exercise 11.6.2: Formulate a reasonable notion of a piecewise monotone function, and then -- show that all bounded piecewise monotone functions are Riemann integrable. -/-- Exercise 11.6.4 -/ +/-- Exercise 11.6.4 (a) -/ example : ∃ (f:ℝ → ℝ), (∀ x ≥ 0, f x ≥ 0) ∧ Summable (fun n:ℕ ↦ f n) ∧ ¬ ∃ M, ∀ N ≥ 0, integ f (Icc 0 N) ≤ M := by sorry +/-- Exercise 11.6.4 (b) -/ example : ∃ (f:ℝ → ℝ), (∀ x ≥ 0, f x ≥ 0) ∧ ¬ Summable (fun n:ℕ ↦ f n) ∧ ∃ M, ∀ N ≥ 0, integ f (Icc 0 N) ≤ M := by sorry diff --git a/Analysis/Section_3_1.lean b/Analysis/Section_3_1.lean index d249d5b4..bd257086 100644 --- a/Analysis/Section_3_1.lean +++ b/Analysis/Section_3_1.lean @@ -731,45 +731,45 @@ example : ({3,5,9}:Set).replace (P := fun _ y ↦ y=1) (by aesop) = {1} := by /-- Exercise 3.1.5. One can use the {tactic}`tfae_have` and {tactic}`tfae_finish` tactics here. -/ theorem SetTheory.Set.subset_tfae (A B:Set) : [A ⊆ B, A ∪ B = B, A ∩ B = A].TFAE := by sorry -/-- Exercise 3.1.7 -/ +/-- Exercise 3.1.7 (a) -/ theorem SetTheory.Set.inter_subset_left (A B:Set) : A ∩ B ⊆ A := by sorry -/-- Exercise 3.1.7 -/ +/-- Exercise 3.1.7 (b) -/ theorem SetTheory.Set.inter_subset_right (A B:Set) : A ∩ B ⊆ B := by sorry -/-- Exercise 3.1.7 -/ +/-- Exercise 3.1.7 (c) -/ @[simp] theorem SetTheory.Set.subset_inter_iff (A B C:Set) : C ⊆ A ∩ B ↔ C ⊆ A ∧ C ⊆ B := by sorry -/-- Exercise 3.1.7 -/ +/-- Exercise 3.1.7 (d) -/ theorem SetTheory.Set.subset_union_left (A B:Set) : A ⊆ A ∪ B := by sorry -/-- Exercise 3.1.7 -/ +/-- Exercise 3.1.7 (e) -/ theorem SetTheory.Set.subset_union_right (A B:Set) : B ⊆ A ∪ B := by sorry -/-- Exercise 3.1.7 -/ +/-- Exercise 3.1.7 (f) -/ @[simp] theorem SetTheory.Set.union_subset_iff (A B C:Set) : A ∪ B ⊆ C ↔ A ⊆ C ∧ B ⊆ C := by sorry -/-- Exercise 3.1.8 -/ +/-- Exercise 3.1.8 (a) -/ @[simp] theorem SetTheory.Set.inter_union_cancel (A B:Set) : A ∩ (A ∪ B) = A := by sorry -/-- Exercise 3.1.8 -/ +/-- Exercise 3.1.8 (b) -/ @[simp] theorem SetTheory.Set.union_inter_cancel (A B:Set) : A ∪ (A ∩ B) = A := by sorry -/-- Exercise 3.1.9 -/ +/-- Exercise 3.1.9 (a) -/ theorem SetTheory.Set.partition_left {A B X:Set} (h_union: A ∪ B = X) (h_inter: A ∩ B = ∅) : A = X \ B := by sorry -/-- Exercise 3.1.9 -/ +/-- Exercise 3.1.9 (b) -/ theorem SetTheory.Set.partition_right {A B X:Set} (h_union: A ∪ B = X) (h_inter: A ∩ B = ∅) : B = X \ A := by sorry