Start fixing up unverified.

2023-08-09 12:59:23 -06:00 · 2023-08-09 12:59:23 -06:00 · 1eff704803
parent e08e68bbc3
commit 1eff704803
3 changed files with 148 additions and 30 deletions
--- a/Bookshelf/Apostol/Chapter_I_03.lean
+++ b/Bookshelf/Apostol/Chapter_I_03.lean
@ -9,6 +9,7 @@ A Set of Axioms for the Real-Number System
 namespace Apostol.Chapter_I_03

 #check Archimedean
+
 #check Real.exists_isLUB

 /-! ## The least-upper-bound axiom (completeness axiom) -/
--- a/Bookshelf/Enderton/Set.tex
+++ b/Bookshelf/Enderton/Set.tex
@ -1165,7 +1165,7 @@

  \end{proof}

-\subsection{\pending{Monotonicity}}%
+\subsection{\verified{Monotonicity}}%
 \hyperlabel{sub:monotonicity}

  For any sets $A$, $B$, and $C$,
@ -1175,11 +1175,23 @@
      A \subseteq B & \Rightarrow \bigcup A \subseteq \bigcup B
    \end{align*}

-  \lean{Mathlib/Data/Set/Basic}{Set.union\_subset\_union\_left}
+  \lean{Mathlib/Data/Set/Basic}
+    {Set.union\_subset\_union\_left}

-  \lean{Mathlib/Data/Set/Basic}{Set.inter\_subset\_inter\_left}
+  \code{Bookshelf/Enderton/Set/Chapter\_2}
+    {Enderton.Set.Chapter\_2.monotonicity\_i}

-  \lean{Mathlib/Data/Set/Lattice}{Set.sUnion\_mono}
+  \lean{Mathlib/Data/Set/Basic}
+    {Set.inter\_subset\_inter\_left}
+
+  \code{Bookshelf/Enderton/Set/Chapter\_2}
+    {Enderton.Set.Chapter\_2.monotonicity\_ii}
+
+  \lean{Mathlib/Data/Set/Lattice}
+    {Set.sUnion\_mono}
+
+  \code{Bookshelf/Enderton/Set/Chapter\_2}
+    {Enderton.Set.Chapter\_2.monotonicity\_iii}

  \begin{proof}

@ -1236,7 +1248,7 @@

  \end{proof}

-\subsection{\pending{Anti-monotonicity}}%
+\subsection{\verified{Anti-monotonicity}}%
 \hyperlabel{sub:anti-monotonicity}

  For any sets $A$, $B$, and $C$,
@ -1245,9 +1257,17 @@
      \emptyset \neq A \subseteq B & \Rightarrow \bigcap B \subseteq \bigcap A.
    \end{align*}

-  \lean{Mathlib/Data/Set/Basic}{Set.diff\_subset\_diff\_right}
+  \lean{Mathlib/Data/Set/Basic}
+    {Set.diff\_subset\_diff\_right}

-  \lean{Mathlib/Data/Set/Lattice}{Set.sInter\_subset\_sInter}
+  \code{Bookshelf/Enderton/Set/Chapter\_2}
+    {Enderton.Set.Chapter\_2.anti\_monotonicity\_i}
+
+  \lean{Mathlib/Data/Set/Lattice}
+    {Set.sInter\_subset\_sInter}
+
+  \code{Bookshelf/Enderton/Set/Chapter\_2}
+    {Enderton.Set.Chapter\_2.anti\_monotonicity\_ii}

  \begin{proof}

@ -1404,14 +1424,18 @@

  \end{proof}

-\subsection{\pending{%
+\subsection{\verified{%
  \texorpdfstring{$\cap$/$-$}{Intersection/Difference} Associativity}}%
 \hyperlabel{sub:intersection-difference-associativity}

  Let $A$, $B$, and $C$ be sets.
  Then $A \cap (B - C) = (A \cap B) - C$.

-  \lean*{Mathlib/Data/Set/Basic}{Set.inter\_diff\_assoc}
+  \lean*{Mathlib/Data/Set/Basic}
+    {Set.inter\_diff\_assoc}
+
+  \code{Bookshelf/Enderton/Set/Chapter\_2}
+    {Enderton.Set.Chapter\_2.inter\_diff\_assoc}

  \begin{proof}
    Let $A$, $B$, and $C$ be sets.
@ -2626,7 +2650,7 @@
    This concludes our proof.
  \end{proof}

-\subsection{\pending{Corollary 3C}}%
+\subsection{\unverified{Corollary 3C}}%
 \hyperlabel{sub:corollary-3c}

  \begin{theorem}[3C]
@ -2634,13 +2658,14 @@
      pairs $\pair{x, y}$ with $x \in A$ and $y \in B$.
  \end{theorem}

+  \begin{note}
+    The below Lean proof is a definition (i.e. an axiom).
+    It does not prove such a set's existence from first principles.
+  \end{note}
+
  \lean{Mathlib/SetTheory/ZFC/Basic}{Set.prod}

  \begin{proof}
-    \begin{note}
-      The above Lean proof is a definition (i.e. an axiom).
-      It does not prove such a set's existence from first principles.
-    \end{note}
    Define $C = A \cup B$.
    Then for all $x \in A$ and for all $y \in B$, $x$ and $y$ are both in $C$.
    By \nameref{sub:lemma-3b}, it follows that
--- a/Bookshelf/Enderton/Set/Chapter_2.lean
+++ b/Bookshelf/Enderton/Set/Chapter_2.lean
@ -18,8 +18,6 @@ A ∩ B = B ∩ A
 ```
 -/

-#check Set.union_comm
-
 theorem commutative_law_i (A B : Set α)
  : A ∪ B = B ∪ A := calc A ∪ B
  _ = { x | x ∈ A ∨ x ∈ B } := rfl
@ -28,7 +26,7 @@ theorem commutative_law_i (A B : Set α)
    exact or_comm
  _ = B ∪ A := rfl

-#check Set.inter_comm
+#check Set.union_comm

 theorem commutative_law_ii (A B : Set α)
  : A ∩ B = B ∩ A := calc A ∩ B
@ -38,6 +36,8 @@ theorem commutative_law_ii (A B : Set α)
    exact and_comm
  _ = B ∩ A := rfl

+#check Set.inter_comm
+
 /-! #### Associative Laws

 For any sets `A`, `B`, and `C`,
@ -47,8 +47,6 @@ A ∩ (B ∩ C) = (A ∩ B) ∩ C
 ```
 -/

-#check Set.union_assoc
-
 theorem associative_law_i (A B C : Set α)
  : A ∪ (B ∪ C) = (A ∪ B) ∪ C := calc A ∪ (B ∪ C)
  _ = { x | x ∈ A ∨ x ∈ B ∪ C } := rfl
@ -60,7 +58,7 @@ theorem associative_law_i (A B C : Set α)
  _ = { x | x ∈ A ∪ B ∨ x ∈ C } := rfl
  _ = (A ∪ B) ∪ C := rfl

-#check Set.inter_assoc
+#check Set.union_assoc

 theorem associative_law_ii (A B C : Set α)
  : A ∩ (B ∩ C) = (A ∩ B) ∩ C := calc A ∩ (B ∩ C)
@ -73,6 +71,8 @@ theorem associative_law_ii (A B C : Set α)
  _ = { x | x ∈ A ∩ B ∧ x ∈ C } := rfl
  _ = (A ∩ B) ∩ C := rfl

+#check Set.inter_assoc
+
 /-! #### Distributive Laws

 For any sets `A`, `B`, and `C`,
@ -82,8 +82,6 @@ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
 ```
 -/

-#check Set.inter_distrib_left
-
 theorem distributive_law_i (A B C : Set α)
  : A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) := calc A ∩ (B ∪ C)
  _ = { x | x ∈ A ∧ x ∈ B ∪ C } := rfl
@ -94,7 +92,7 @@ theorem distributive_law_i (A B C : Set α)
  _ = { x | x ∈ A ∩ B ∨ x ∈ A ∩ C } := rfl
  _ = (A ∩ B) ∪ (A ∩ C) := rfl

-#check Set.union_distrib_left
+#check Set.inter_distrib_left

 theorem distributive_law_ii (A B C : Set α)
  : A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) := calc A ∪ (B ∩ C)
@ -106,6 +104,8 @@ theorem distributive_law_ii (A B C : Set α)
  _ = { x | x ∈ A ∪ B ∧ x ∈ A ∪ C } := rfl
  _ = (A ∪ B) ∩ (A ∪ C) := rfl

+#check Set.union_distrib_left
+
 /-! #### De Morgan's Laws

 For any sets `A`, `B`, and `C`,
@ -115,8 +115,6 @@ C - (A ∩ B) = (C - A) ∪ (C - B)
 ```
 -/

-#check Set.diff_inter_diff
-
 theorem de_morgans_law_i (A B C : Set α)
  : C \ (A ∪ B) = (C \ A) ∩ (C \ B) := calc C \ (A ∪ B)
  _ = { x | x ∈ C ∧ x ∉ A ∪ B } := rfl
@ -131,7 +129,7 @@ theorem de_morgans_law_i (A B C : Set α)
  _ = { x | x ∈ C \ A ∧ x ∈ C \ B } := rfl
  _ = (C \ A) ∩ (C \ B) := rfl

-#check Set.diff_inter
+#check Set.diff_inter_diff

 theorem de_morgans_law_ii (A B C : Set α)
  : C \ (A ∩ B) = (C \ A) ∪ (C \ B) := calc C \ (A ∩ B)
@ -147,6 +145,8 @@ theorem de_morgans_law_ii (A B C : Set α)
  _ = { x | x ∈ C \ A ∨ x ∈ C \ B } := rfl
  _ = (C \ A) ∪ (C \ B) := rfl

+#check Set.diff_inter
+
 /-! #### Identities Involving ∅

 For any set `A`,
@ -157,8 +157,6 @@ A ∩ (C - A) = ∅
 ```
 -/

-#check Set.union_empty
-
 theorem emptyset_identity_i (A : Set α)
  : A ∪ ∅ = A := calc A ∪ ∅
  _ = { x | x ∈ A ∨ x ∈ ∅ } := rfl
@ -166,7 +164,7 @@ theorem emptyset_identity_i (A : Set α)
  _ = { x | x ∈ A } := by simp
  _ = A := rfl

-#check Set.inter_empty
+#check Set.union_empty

 theorem emptyset_identity_ii (A : Set α)
  : A ∩ ∅ = ∅ := calc A ∩ ∅
@ -175,7 +173,7 @@ theorem emptyset_identity_ii (A : Set α)
  _ = { x | False } := by simp
  _ = ∅ := rfl

-#check Set.inter_diff_self
+#check Set.inter_empty

 theorem emptyset_identity_iii (A C : Set α)
  : A ∩ (C \ A) = ∅ := calc A ∩ (C \ A)
@ -185,6 +183,100 @@ theorem emptyset_identity_iii (A C : Set α)
  _ = { x | False } := by simp
  _ = ∅ := rfl

+#check Set.inter_diff_self
+
+/-! #### Monotonicity
+
+For any sets `A`, `B`, and `C`,
+```
+A ⊆ B ⇒ A ∪ C ⊆ B ∪ C
+A ⊆ B ⇒ A ∩ C ⊆ B ∩ C
+A ⊆ B ⇒ ⋃ A ⊆ ⋃ B
+```
+-/
+
+theorem monotonicity_i (A B C : Set α) (h : A ⊆ B)
+  : A ∪ C ⊆ B ∪ C := by
+  show ∀ x, x ∈ A ∪ C → x ∈ B ∪ C
+  intro x hx
+  apply Or.elim hx
+  · intro hA
+    have := h hA
+    left
+    exact this
+  · intro hC
+    right
+    exact hC
+
+#check Set.union_subset_union_left
+
+theorem monotonicity_ii (A B C : Set α) (h : A ⊆ B)
+  : A ∩ C ⊆ B ∩ C := by
+  show ∀ x, x ∈ A ∩ C → x ∈ B ∩ C
+  intro x hx
+  have := h hx.left
+  exact ⟨this, hx.right⟩
+
+#check Set.inter_subset_inter_left
+
+theorem monotonicity_iii (A B : Set (Set α)) (h : A ⊆ B)
+  : ⋃₀ A ⊆ ⋃₀ B := by
+  show ∀ x, x ∈ ⋃₀ A → x ∈ ⋃₀ B
+  intro x hx
+  have ⟨b, hb⟩ := hx
+  have := h hb.left
+  exact ⟨b, this, hb.right⟩
+
+#check Set.sUnion_mono
+
+/-! #### Anti-monotonicity
+
+For any sets `A`, `B`, and `C`,
+```
+A ⊆ B ⇒ C - B ⊆ C - A
+∅ ≠ A ⊆ B ⇒ ⋂ B ⊆ ⋂ A
+```
+-/
+
+theorem anti_monotonicity_i (A B C : Set α) (h : A ⊆ B)
+  : C \ B ⊆ C \ A := by
+  show ∀ x, x ∈ C \ B → x ∈ C \ A
+  intro x hx
+  have : x ∉ A := by
+    by_contra nh
+    have := h nh
+    exact absurd this hx.right
+  exact ⟨hx.left, this⟩
+
+#check Set.diff_subset_diff_right
+
+theorem anti_monotonicity_ii (A B : Set (Set α)) (h : A ⊆ B)
+  : ⋂₀ B ⊆ ⋂₀ A := by
+  show ∀ x, x ∈ ⋂₀ B → x ∈ ⋂₀ A
+  intro x hx
+  have : ∀ b, b ∈ B → x ∈ b := hx
+  show ∀ a, a ∈ A → x ∈ a
+  intro a ha
+  exact this a (h ha)
+
+#check Set.sInter_subset_sInter
+
+/-- #### ∩/- Associativity
+
+Let `A`, `B`, and `C` be sets. Then `A ∩ (B - C) = (A ∩ B) - C`.
+-/
+theorem inter_diff_assoc (A B C : Set α)
+  : A ∩ (B \ C) = (A ∩ B) \ C := calc A ∩ (B \ C)
+  _ = { x | x ∈ A ∧ x ∈ (B \ C) } := rfl
+  _ = { x | x ∈ A ∧ (x ∈ B ∧ x ∉ C) } := rfl
+  _ = { x | (x ∈ A ∧ x ∈ B) ∧ x ∉ C } := by
+    ext _
+    sorry
+  _ = { x | x ∈ A ∩ B ∧ x ∉ C } := rfl
+  _ = (A ∩ B) \ C := rfl
+
+#check Set.inter_diff_assoc
+
 /-- #### Exercise 2.1

 Assume that `A` is the set of integers divisible by `4`. Similarly assume that