Fix build errors in `Chapter_2.lean`.

finite-set-exercises
Joshua Potter 2023-06-10 12:09:24 -06:00
parent 64d8324657
commit 6d3a2d8ad0
3 changed files with 9 additions and 13 deletions

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@ -91,7 +91,7 @@ theorem exercise_3_3 {A : Set (Set α)}
Show that if `A ⊆ B`, then ` A ⊆ B`.
-/
theorem exercise_3_4 (h : A ⊆ B) : ⋃₀ A ⊆ ⋃₀ B := by
theorem exercise_3_4 {A B : Set (Set α)} (h : A ⊆ B) : ⋃₀ A ⊆ ⋃₀ B := by
show ∀ x ∈ { a | ∃ t, t ∈ A ∧ a ∈ t }, x ∈ { a | ∃ t, t ∈ B ∧ a ∈ t }
intro x hx
rw [Set.mem_setOf_eq] at hx
@ -103,7 +103,8 @@ theorem exercise_3_4 (h : A ⊆ B) : ⋃₀ A ⊆ ⋃₀ B := by
Assume that every member of `𝓐` is a subset of `B`. Show that ` 𝓐 ⊆ B`.
-/
theorem exercise_3_5 (h : ∀ x ∈ 𝓐, x ⊆ B) : ⋃₀ 𝓐 ⊆ B := by
theorem exercise_3_5 {𝓐 : Set (Set α)} (h : ∀ x ∈ 𝓐, x ⊆ B)
: ⋃₀ 𝓐 ⊆ B := by
show ∀ y ∈ { a | ∃ t, t ∈ 𝓐 ∧ a ∈ t }, y ∈ B
intro y hy
rw [Set.mem_setOf_eq] at hy
@ -267,7 +268,7 @@ theorem exercise_3_9 (ha : a = {1}) (hB : B = {{1}})
Show that if `a ∈ B`, then `𝓟 a ∈ 𝓟 𝓟 B`.
-/
theorem exercise_3_10 (ha : a ∈ B)
theorem exercise_3_10 {B : Set (Set α)} (ha : a ∈ B)
: 𝒫 a ∈ 𝒫 (𝒫 (⋃₀ B)) := by
have h₁ := exercise_3_3 a ha
have h₂ := Chapter_1.exercise_1_3 h₁

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@ -115,14 +115,4 @@ theorem exercise_5_3 {A : Set (Set α)} {𝓑 : Set (Set β)}
· intro ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
exact ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
/-- ### Exercise 5.5b
With `A`, `B`, and `C` as above, show that `A × B = C`.
-/
theorem exercise_5_5b {A : Set α} {B : Set β}
: Set.prod A B = ⋃₀ {Set.prod {x} B | x ∈ A} := by
-- TODO: `Set.OrderedPair` should allow two different types.
-- TODO: We can cast `(α × β)` up into type `Set (Set (α ⊕ β))`.
sorry
end Enderton.Set.Chapter_3

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@ -7,11 +7,16 @@ namespace Set
/--
Kazimierz Kuratowski's definition of an ordered pair.
Like `Set`, this is a homogeneous structure.
-/
def OrderedPair (x y : α) : Set (Set α) := {{x}, {x, y}}
namespace OrderedPair
/--
For any sets `x`, `y`, `u`, and `v`, `⟨u, v⟩ = ⟨x, y⟩` **iff** `u = x ∧ v = y`.
-/
theorem ext_iff
: (OrderedPair x y = OrderedPair u v) ↔ (x = u ∧ y = v) := by
unfold OrderedPair