Start fixing up unverified.
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@ -9,6 +9,7 @@ A Set of Axioms for the Real-Number System
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namespace Apostol.Chapter_I_03
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namespace Apostol.Chapter_I_03
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#check Archimedean
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#check Archimedean
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#check Real.exists_isLUB
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#check Real.exists_isLUB
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/-! ## The least-upper-bound axiom (completeness axiom) -/
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/-! ## The least-upper-bound axiom (completeness axiom) -/
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@ -1165,7 +1165,7 @@
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\end{proof}
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\end{proof}
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\subsection{\pending{Monotonicity}}%
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\subsection{\verified{Monotonicity}}%
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\hyperlabel{sub:monotonicity}
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\hyperlabel{sub:monotonicity}
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For any sets $A$, $B$, and $C$,
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For any sets $A$, $B$, and $C$,
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@ -1175,11 +1175,23 @@
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A \subseteq B & \Rightarrow \bigcup A \subseteq \bigcup B
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A \subseteq B & \Rightarrow \bigcup A \subseteq \bigcup B
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\end{align*}
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\end{align*}
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\lean{Mathlib/Data/Set/Basic}{Set.union\_subset\_union\_left}
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\lean{Mathlib/Data/Set/Basic}
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{Set.union\_subset\_union\_left}
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\lean{Mathlib/Data/Set/Basic}{Set.inter\_subset\_inter\_left}
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\code{Bookshelf/Enderton/Set/Chapter\_2}
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{Enderton.Set.Chapter\_2.monotonicity\_i}
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\lean{Mathlib/Data/Set/Lattice}{Set.sUnion\_mono}
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\lean{Mathlib/Data/Set/Basic}
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{Set.inter\_subset\_inter\_left}
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\code{Bookshelf/Enderton/Set/Chapter\_2}
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{Enderton.Set.Chapter\_2.monotonicity\_ii}
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\lean{Mathlib/Data/Set/Lattice}
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{Set.sUnion\_mono}
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\code{Bookshelf/Enderton/Set/Chapter\_2}
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{Enderton.Set.Chapter\_2.monotonicity\_iii}
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\begin{proof}
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\begin{proof}
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@ -1236,7 +1248,7 @@
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\end{proof}
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\end{proof}
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\subsection{\pending{Anti-monotonicity}}%
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\subsection{\verified{Anti-monotonicity}}%
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\hyperlabel{sub:anti-monotonicity}
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\hyperlabel{sub:anti-monotonicity}
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For any sets $A$, $B$, and $C$,
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For any sets $A$, $B$, and $C$,
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@ -1245,9 +1257,17 @@
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\emptyset \neq A \subseteq B & \Rightarrow \bigcap B \subseteq \bigcap A.
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\emptyset \neq A \subseteq B & \Rightarrow \bigcap B \subseteq \bigcap A.
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\end{align*}
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\end{align*}
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\lean{Mathlib/Data/Set/Basic}{Set.diff\_subset\_diff\_right}
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\lean{Mathlib/Data/Set/Basic}
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{Set.diff\_subset\_diff\_right}
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\lean{Mathlib/Data/Set/Lattice}{Set.sInter\_subset\_sInter}
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\code{Bookshelf/Enderton/Set/Chapter\_2}
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{Enderton.Set.Chapter\_2.anti\_monotonicity\_i}
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\lean{Mathlib/Data/Set/Lattice}
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{Set.sInter\_subset\_sInter}
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\code{Bookshelf/Enderton/Set/Chapter\_2}
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{Enderton.Set.Chapter\_2.anti\_monotonicity\_ii}
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\begin{proof}
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\begin{proof}
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@ -1404,14 +1424,18 @@
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\end{proof}
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\end{proof}
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\subsection{\pending{%
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\subsection{\verified{%
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\texorpdfstring{$\cap$/$-$}{Intersection/Difference} Associativity}}%
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\texorpdfstring{$\cap$/$-$}{Intersection/Difference} Associativity}}%
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\hyperlabel{sub:intersection-difference-associativity}
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\hyperlabel{sub:intersection-difference-associativity}
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Let $A$, $B$, and $C$ be sets.
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Let $A$, $B$, and $C$ be sets.
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Then $A \cap (B - C) = (A \cap B) - C$.
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Then $A \cap (B - C) = (A \cap B) - C$.
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\lean*{Mathlib/Data/Set/Basic}{Set.inter\_diff\_assoc}
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\lean*{Mathlib/Data/Set/Basic}
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{Set.inter\_diff\_assoc}
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\code{Bookshelf/Enderton/Set/Chapter\_2}
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{Enderton.Set.Chapter\_2.inter\_diff\_assoc}
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\begin{proof}
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\begin{proof}
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Let $A$, $B$, and $C$ be sets.
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Let $A$, $B$, and $C$ be sets.
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@ -2626,7 +2650,7 @@
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This concludes our proof.
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This concludes our proof.
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\end{proof}
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\end{proof}
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\subsection{\pending{Corollary 3C}}%
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\subsection{\unverified{Corollary 3C}}%
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\hyperlabel{sub:corollary-3c}
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\hyperlabel{sub:corollary-3c}
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\begin{theorem}[3C]
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\begin{theorem}[3C]
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@ -2634,13 +2658,14 @@
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pairs $\pair{x, y}$ with $x \in A$ and $y \in B$.
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pairs $\pair{x, y}$ with $x \in A$ and $y \in B$.
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\end{theorem}
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\end{theorem}
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\begin{note}
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The below Lean proof is a definition (i.e. an axiom).
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It does not prove such a set's existence from first principles.
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\end{note}
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\lean{Mathlib/SetTheory/ZFC/Basic}{Set.prod}
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\lean{Mathlib/SetTheory/ZFC/Basic}{Set.prod}
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\begin{proof}
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\begin{proof}
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\begin{note}
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The above Lean proof is a definition (i.e. an axiom).
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It does not prove such a set's existence from first principles.
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\end{note}
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Define $C = A \cup B$.
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Define $C = A \cup B$.
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Then for all $x \in A$ and for all $y \in B$, $x$ and $y$ are both in $C$.
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Then for all $x \in A$ and for all $y \in B$, $x$ and $y$ are both in $C$.
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By \nameref{sub:lemma-3b}, it follows that
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By \nameref{sub:lemma-3b}, it follows that
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@ -18,8 +18,6 @@ A ∩ B = B ∩ A
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```
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```
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-/
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-/
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#check Set.union_comm
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theorem commutative_law_i (A B : Set α)
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theorem commutative_law_i (A B : Set α)
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: A ∪ B = B ∪ A := calc A ∪ B
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: A ∪ B = B ∪ A := calc A ∪ B
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_ = { x | x ∈ A ∨ x ∈ B } := rfl
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_ = { x | x ∈ A ∨ x ∈ B } := rfl
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@ -28,7 +26,7 @@ theorem commutative_law_i (A B : Set α)
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exact or_comm
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exact or_comm
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_ = B ∪ A := rfl
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_ = B ∪ A := rfl
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#check Set.inter_comm
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#check Set.union_comm
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theorem commutative_law_ii (A B : Set α)
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theorem commutative_law_ii (A B : Set α)
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: A ∩ B = B ∩ A := calc A ∩ B
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: A ∩ B = B ∩ A := calc A ∩ B
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@ -38,6 +36,8 @@ theorem commutative_law_ii (A B : Set α)
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exact and_comm
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exact and_comm
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_ = B ∩ A := rfl
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_ = B ∩ A := rfl
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#check Set.inter_comm
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/-! #### Associative Laws
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/-! #### Associative Laws
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For any sets `A`, `B`, and `C`,
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For any sets `A`, `B`, and `C`,
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@ -47,8 +47,6 @@ A ∩ (B ∩ C) = (A ∩ B) ∩ C
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```
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```
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-/
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-/
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#check Set.union_assoc
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theorem associative_law_i (A B C : Set α)
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theorem associative_law_i (A B C : Set α)
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: A ∪ (B ∪ C) = (A ∪ B) ∪ C := calc A ∪ (B ∪ C)
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: A ∪ (B ∪ C) = (A ∪ B) ∪ C := calc A ∪ (B ∪ C)
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_ = { x | x ∈ A ∨ x ∈ B ∪ C } := rfl
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_ = { x | x ∈ A ∨ x ∈ B ∪ C } := rfl
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@ -60,7 +58,7 @@ theorem associative_law_i (A B C : Set α)
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_ = { x | x ∈ A ∪ B ∨ x ∈ C } := rfl
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_ = { x | x ∈ A ∪ B ∨ x ∈ C } := rfl
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_ = (A ∪ B) ∪ C := rfl
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_ = (A ∪ B) ∪ C := rfl
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#check Set.inter_assoc
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#check Set.union_assoc
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theorem associative_law_ii (A B C : Set α)
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theorem associative_law_ii (A B C : Set α)
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: A ∩ (B ∩ C) = (A ∩ B) ∩ C := calc A ∩ (B ∩ C)
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: A ∩ (B ∩ C) = (A ∩ B) ∩ C := calc A ∩ (B ∩ C)
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_ = { x | x ∈ A ∩ B ∧ x ∈ C } := rfl
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_ = { x | x ∈ A ∩ B ∧ x ∈ C } := rfl
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_ = (A ∩ B) ∩ C := rfl
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_ = (A ∩ B) ∩ C := rfl
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#check Set.inter_assoc
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/-! #### Distributive Laws
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/-! #### Distributive Laws
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For any sets `A`, `B`, and `C`,
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For any sets `A`, `B`, and `C`,
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```
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```
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-/
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-/
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#check Set.inter_distrib_left
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theorem distributive_law_i (A B C : Set α)
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theorem distributive_law_i (A B C : Set α)
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: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) := calc A ∩ (B ∪ C)
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: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) := calc A ∩ (B ∪ C)
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_ = { x | x ∈ A ∧ x ∈ B ∪ C } := rfl
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_ = { x | x ∈ A ∧ x ∈ B ∪ C } := rfl
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_ = { x | x ∈ A ∩ B ∨ x ∈ A ∩ C } := rfl
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_ = { x | x ∈ A ∩ B ∨ x ∈ A ∩ C } := rfl
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_ = (A ∩ B) ∪ (A ∩ C) := rfl
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_ = (A ∩ B) ∪ (A ∩ C) := rfl
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#check Set.union_distrib_left
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#check Set.inter_distrib_left
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theorem distributive_law_ii (A B C : Set α)
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theorem distributive_law_ii (A B C : Set α)
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: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) := calc A ∪ (B ∩ C)
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: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) := calc A ∪ (B ∩ C)
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_ = { x | x ∈ A ∪ B ∧ x ∈ A ∪ C } := rfl
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_ = { x | x ∈ A ∪ B ∧ x ∈ A ∪ C } := rfl
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_ = (A ∪ B) ∩ (A ∪ C) := rfl
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_ = (A ∪ B) ∩ (A ∪ C) := rfl
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#check Set.union_distrib_left
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/-! #### De Morgan's Laws
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/-! #### De Morgan's Laws
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For any sets `A`, `B`, and `C`,
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For any sets `A`, `B`, and `C`,
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```
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```
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-/
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-/
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#check Set.diff_inter_diff
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theorem de_morgans_law_i (A B C : Set α)
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theorem de_morgans_law_i (A B C : Set α)
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: C \ (A ∪ B) = (C \ A) ∩ (C \ B) := calc C \ (A ∪ B)
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: C \ (A ∪ B) = (C \ A) ∩ (C \ B) := calc C \ (A ∪ B)
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_ = { x | x ∈ C ∧ x ∉ A ∪ B } := rfl
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_ = { x | x ∈ C ∧ x ∉ A ∪ B } := rfl
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_ = { x | x ∈ C \ A ∧ x ∈ C \ B } := rfl
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_ = { x | x ∈ C \ A ∧ x ∈ C \ B } := rfl
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_ = (C \ A) ∩ (C \ B) := rfl
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_ = (C \ A) ∩ (C \ B) := rfl
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#check Set.diff_inter
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#check Set.diff_inter_diff
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theorem de_morgans_law_ii (A B C : Set α)
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theorem de_morgans_law_ii (A B C : Set α)
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: C \ (A ∩ B) = (C \ A) ∪ (C \ B) := calc C \ (A ∩ B)
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: C \ (A ∩ B) = (C \ A) ∪ (C \ B) := calc C \ (A ∩ B)
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_ = { x | x ∈ C \ A ∨ x ∈ C \ B } := rfl
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_ = { x | x ∈ C \ A ∨ x ∈ C \ B } := rfl
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_ = (C \ A) ∪ (C \ B) := rfl
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_ = (C \ A) ∪ (C \ B) := rfl
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#check Set.diff_inter
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/-! #### Identities Involving ∅
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/-! #### Identities Involving ∅
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For any set `A`,
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For any set `A`,
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```
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```
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-/
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-/
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#check Set.union_empty
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theorem emptyset_identity_i (A : Set α)
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theorem emptyset_identity_i (A : Set α)
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: A ∪ ∅ = A := calc A ∪ ∅
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: A ∪ ∅ = A := calc A ∪ ∅
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_ = { x | x ∈ A ∨ x ∈ ∅ } := rfl
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_ = { x | x ∈ A ∨ x ∈ ∅ } := rfl
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_ = { x | x ∈ A } := by simp
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_ = { x | x ∈ A } := by simp
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_ = A := rfl
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_ = A := rfl
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#check Set.inter_empty
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#check Set.union_empty
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theorem emptyset_identity_ii (A : Set α)
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theorem emptyset_identity_ii (A : Set α)
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: A ∩ ∅ = ∅ := calc A ∩ ∅
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: A ∩ ∅ = ∅ := calc A ∩ ∅
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_ = { x | False } := by simp
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_ = { x | False } := by simp
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_ = ∅ := rfl
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_ = ∅ := rfl
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#check Set.inter_diff_self
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#check Set.inter_empty
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theorem emptyset_identity_iii (A C : Set α)
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theorem emptyset_identity_iii (A C : Set α)
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: A ∩ (C \ A) = ∅ := calc A ∩ (C \ A)
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: A ∩ (C \ A) = ∅ := calc A ∩ (C \ A)
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_ = { x | False } := by simp
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_ = { x | False } := by simp
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_ = ∅ := rfl
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_ = ∅ := rfl
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#check Set.inter_diff_self
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/-! #### Monotonicity
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For any sets `A`, `B`, and `C`,
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```
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A ⊆ B ⇒ A ∪ C ⊆ B ∪ C
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A ⊆ B ⇒ A ∩ C ⊆ B ∩ C
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A ⊆ B ⇒ ⋃ A ⊆ ⋃ B
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```
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-/
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theorem monotonicity_i (A B C : Set α) (h : A ⊆ B)
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: A ∪ C ⊆ B ∪ C := by
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show ∀ x, x ∈ A ∪ C → x ∈ B ∪ C
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intro x hx
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apply Or.elim hx
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· intro hA
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have := h hA
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left
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exact this
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· intro hC
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right
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exact hC
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#check Set.union_subset_union_left
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theorem monotonicity_ii (A B C : Set α) (h : A ⊆ B)
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: A ∩ C ⊆ B ∩ C := by
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show ∀ x, x ∈ A ∩ C → x ∈ B ∩ C
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intro x hx
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have := h hx.left
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exact ⟨this, hx.right⟩
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#check Set.inter_subset_inter_left
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theorem monotonicity_iii (A B : Set (Set α)) (h : A ⊆ B)
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: ⋃₀ A ⊆ ⋃₀ B := by
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show ∀ x, x ∈ ⋃₀ A → x ∈ ⋃₀ B
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|
intro x hx
|
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|
have ⟨b, hb⟩ := hx
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|
have := h hb.left
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|
exact ⟨b, this, hb.right⟩
|
||||||
|
|
||||||
|
#check Set.sUnion_mono
|
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|
|
||||||
|
/-! #### 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
|
/-- #### Exercise 2.1
|
||||||
|
|
||||||
Assume that `A` is the set of integers divisible by `4`. Similarly assume that
|
Assume that `A` is the set of integers divisible by `4`. Similarly assume that
|
||||||
|
|
Loading…
Reference in New Issue