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4 changes: 2 additions & 2 deletions Analysis/Section_3_5.lean
Original file line number Diff line number Diff line change
Expand Up @@ -439,13 +439,13 @@ theorem SetTheory.Set.diff_prod (A B C:Set) : (A \ B) ×ˢ C = (A ×ˢ C) \ (B
theorem SetTheory.Set.inter_of_prod (A B C D:Set) :
(A ×ˢ B) ∩ (C ×ˢ D) = (A ∩ C) ×ˢ (B ∩ D) := by sorry

/- Exercise 3.5.5 -/
/-- Exercise 3.5.5 -/
def SetTheory.Set.union_of_prod :
Decidable (∀ (A B C D:Set), (A ×ˢ B) ∪ (C ×ˢ D) = (A ∪ C) ×ˢ (B ∪ D)) := by
-- the first line of this construction should be `apply isTrue` or `apply isFalse`.
sorry

/- Exercise 3.5.5 -/
/-- Exercise 3.5.5 -/
def SetTheory.Set.diff_of_prod :
Decidable (∀ (A B C D:Set), (A ×ˢ B) \ (C ×ˢ D) = (A \ C) ×ˢ (B \ D)) := by
-- the first line of this construction should be `apply isTrue` or `apply isFalse`.
Expand Down
2 changes: 1 addition & 1 deletion Analysis/Section_6_4.lean
Original file line number Diff line number Diff line change
Expand Up @@ -308,7 +308,7 @@ theorem Sequence.Cauchy_iff_convergent (a:Sequence) :
theorem Sequence.sup_not_strict_mono : ∃ (a b:ℕ → ℝ), (∀ n, a n < b n) ∧ ¬ (a:Sequence).sup < (b:Sequence).sup := by
sorry

/- Exercise 6.4.7 -/
/-- Exercise 6.4.7 -/
def Sequence.tendsTo_real_iff :
Decidable (∀ (a:Sequence) (x:ℝ), a.TendsTo x ↔ a.abs.TendsTo x) := by
-- The first line of this construction should be `apply isTrue` or `apply isFalse`.
Expand Down
5 changes: 2 additions & 3 deletions Analysis/Section_6_6.lean
Original file line number Diff line number Diff line change
Expand Up @@ -20,7 +20,7 @@ namespace Chapter6
/-- Definition 6.6.1 -/
abbrev Sequence.subseq (a b: ℕ → ℝ) : Prop := ∃ f : ℕ → ℕ, StrictMono f ∧ ∀ n, b n = a (f n)

/- Example 6.6.2 -/
/-- Example 6.6.2 -/
example (a:ℕ → ℝ) : Sequence.subseq a (fun n ↦ a (2 * n)) := by sorry

example {f: ℕ → ℕ} (hf: StrictMono f) : Function.Injective f := by sorry
Expand Down Expand Up @@ -60,8 +60,7 @@ theorem Sequence.convergent_of_subseq_of_bounded {a:ℕ→ ℝ} (ha: (a:Sequence
have := limit_point_of_limsup hL_plus
rw [limit_point_iff_subseq] at this; peel 2 this; solve_by_elim

/- Exercise 6.6.2 -/

/-- Exercise 6.6.2 -/
def Sequence.exist_subseq_of_subseq :
Decidable (∃ a b : ℕ → ℝ, a ≠ b ∧ Sequence.subseq a b ∧ Sequence.subseq b a) := by
-- The first line of this construction should be `apply isTrue` or `apply isFalse`.
Expand Down
3 changes: 1 addition & 2 deletions Analysis/Section_7_2.lean
Original file line number Diff line number Diff line change
Expand Up @@ -273,8 +273,7 @@ theorem Series.telescope {a:ℕ → ℝ} (ha: Filter.atTop.Tendsto a (nhds 0)) :
((fun n:ℕ ↦ a n - a (n+1)):Series).convergesTo (a 0) := by
sorry

/- Exercise 7.2.1 -/

/-- Exercise 7.2.1 -/
def Series.exercise_7_2_1_convergent :
Decidable ( (mk' (m := 1) (fun n ↦ (-1:ℝ)^(n:ℤ))).converges ) := by
-- The first line of this proof should be `apply isTrue` or `apply isFalse`.
Expand Down
4 changes: 2 additions & 2 deletions Analysis/Section_9_2.lean
Original file line number Diff line number Diff line change
Expand Up @@ -51,12 +51,12 @@ example : f_9_2_2 ∘ g_9_2_2 = fun x ↦ 4*x^2 := by grind

example : g_9_2_2 ∘ f_9_2_2 = fun x ↦ 2*x^2 := by grind

/- Exercise 9.2.1. -/

/-- Exercise 9.2.1. -/
def Exercise_9_2_1a : Decidable (∀ (f g h : ℝ → ℝ), (f+g) ∘ h = f ∘ h + g ∘ h) := by
-- The first line of this construction should be `apply isTrue` or `apply isFalse`.
sorry

/-- Exercise 9.2.1. -/
def Exercise_9_2_1b : Decidable (∀ (f g h : ℝ → ℝ), f ∘ (g + h) = f ∘ g + f ∘ h) := by
-- The first line of this construction should be `apply isTrue` or `apply isFalse`.
sorry
Expand Down
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