Enderton (logic). Begin writing out Lean proofs for Exercises 1.
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import Bookshelf.Enderton.Logic.Chapter_0
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#check Iff
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import Bookshelf.Enderton.Logic.Chapter_1
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@ -62,13 +62,20 @@
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& = (\alpha \Leftrightarrow \beta)
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\end{align*}
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\lean{Init/Prelude}{Not}
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\code*{Bookshelf/Enderton/Logic/Chapter\_1}
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{Enderton.Logic.Chapter\_1.WFF}
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\lean{Init/Prelude}{And}
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\lean{Init/Prelude}
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{Not}
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\lean{Init/Prelude}{Or}
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\lean{Init/Prelude}
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{And}
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\lean{Init/Core}{Iff}
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\lean{Init/Prelude}
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{Or}
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\lean{Init/Core}
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{Iff}
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\section{\defined{\texorpdfstring{$n$}{n}-tuple}}%
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\hyperlabel{ref:n-tuple}
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@ -78,7 +85,8 @@
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for $n > 1$.
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We also define $\tuple{x} = x$.
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\lean*{Init/Prelude}{Prod}
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\lean*{Init/Prelude}
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{Prod}
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\section{\defined{Well-Formed Formula}}%
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\hyperlabel{ref:well-formed-formula}
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@ -87,6 +95,9 @@
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be built up from the sentence symbols by applying some finite number of
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times the \nameref{ref:formula-building-operations}.
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\code*{Bookshelf/Enderton/Logic/Chapter\_1}
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{Enderton.Logic.Chapter\_1.WFF}
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\endgroup
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% Reset counter to mirror Enderton's book.
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@ -285,12 +296,18 @@
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\end{answer}
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\subsection{\unverified{Exercise 1.1.2}}%
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\subsection{\pending{Exercise 1.1.2}}%
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\hyperlabel{sub:exercise-1.1.2}
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Show that there are no wffs of length 2, 3, or 6, but that any other positive
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length is possible.
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\code*{Enderton.Logic.Chapter\_1}
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{Enderton.Logic.Chapter\_1.exercise\_1\_1\_2\_i}
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\code{Enderton.Logic.Chapter\_1}
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{Enderton.Logic.Chapter\_1.exercise\_1\_1\_2\_ii}
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\begin{proof}
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Define $$S = \{ \phi \mid
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@ -351,7 +368,7 @@
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\end{proof}
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\subsection{\sorry{Exercise 1.1.3}}%
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\subsection{\pending{Exercise 1.1.3}}%
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\hyperlabel{sub:exercise-1.1.3}
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Let $\alpha$ be a wff; let $c$ be the number of places at which binary
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@ -363,10 +380,64 @@
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Show by using the induction principle that $s = c + 1$.
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\begin{proof}
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TODO
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Define
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\begin{equation}
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\hyperlabel{sub:exercise-1.1.3-eq1}
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S = \{\phi \mid \phi \text{ is a wff such that } s = c + 1\}.
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\end{equation}
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We prove that (i) all the sentence symbols are members of $S$ and (ii)
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$S$ is closed under the five \nameref{ref:formula-building-operations}.
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We then conclude with (iii) the proof of the theorem statement.
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\paragraph{(i)}%
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\hyperlabel{par:exercise-1.1.3-i}
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Let $\phi = A_n$ be an arbitrary sentence symbol.
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The number of places at which sentence symbols occur in $\phi$ is 1.
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The number of places at which binary connective symbols occur in $\phi$ is
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0.
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Hence $\phi \in S$.
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\paragraph{(ii)}%
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\hyperlabel{par:exercise-1.1.3-ii}
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Let $\alpha, \beta \in S$.
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Denote the number of places at which sentence symbols occur in each as
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$s_\alpha$ and $s_\beta$ respectively.
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Likewise, denote the number of places at which binary connective symbols
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occur as $c_\alpha$ and $c_\beta$.
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By definition, $\mathcal{E}_{\neg}(\alpha) = (\neg\alpha)$.
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The number of sentence and binary connective symbols in
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$\mathcal{E}_{\neg}(\alpha)$ does not change.
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Thus $\mathcal{E}_{\neg}(\alpha) \in S$.
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Likewise,
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$\mathcal{E}_{\square}(\phi, \psi) = (\phi \mathop{\square} \psi)$
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where $\square$ is one of the binary connectives $\land$, $\lor$,
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$\Rightarrow$, $\Leftrightarrow$.
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Therefore $\mathcal{E}_{\square}(\phi, \psi)$ has $s_\alpha + s_\beta$
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sentence symbols and $c_\alpha + c_\beta + 1$ binary connective symbols.
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But \eqref{sub:exercise-1.1.3-eq1} implies
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\begin{align*}
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s_\alpha + s_\beta
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& = (c_\alpha + 1) + (c_\beta + 1) \\
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& = (c_\alpha + c_\beta + 1) + 1,
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\end{align*}
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meaning $\mathcal{E}_{\square}(\phi, \psi) \in S$.
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Hence $S$ is closed under the five formula-building operations.
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\paragraph{(iii)}%
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\hyperlabel{par:exercise-1.1.3-iii}
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By \nameref{par:exercise-1.1.3-i} and \nameref{par:exercise-1.1.3-ii}, the
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\nameref{sub:induction-principle-1} indicates $S$ is the set of all
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wffs.
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\end{proof}
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\subsection{\sorry{Exercise 1.1.4}}%
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\subsection{\unverified{Exercise 1.1.4}}%
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\hyperlabel{sub:exercise-1.1.4}
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Assume we have a construction sequence ending in $\phi$, where $\phi$ does not
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@ -376,20 +447,75 @@
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Show that the result is still a legal construction sequence.
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\begin{proof}
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TODO
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Let $S$ denote a \nameref{ref:construction-sequence}
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$\ltuple{\epsilon_1}{\epsilon_n}$ such that $\epsilon_n = \phi$.
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Let $S' = \ltuple{\epsilon_{i_1}}{\epsilon_{i_m}}$ denote the construction
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sequence resulting from deleting all expressions in $S$ containing $A_4$.
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Fix $1 \leq j \leq m$.
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Then there exists some $1 \leq k \leq n$ such that
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$\epsilon_{i_j} = \epsilon_k$.
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By definition of a construction sequence, there are three cases to consider:
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\paragraph{Case 1}%
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Suppose $\epsilon_k$ is a sentence symbol.
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Then $\epsilon_{i_j}$ is also sentence symbol.
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\paragraph{Case 2}%
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Suppose $\epsilon_k = \mathcal{E}_{\neg}(\epsilon_a)$ for some $a < k$.
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It must be that $A_4$ is not found in $\epsilon_a$, else an immediate
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contradiction is raised.
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Therefore $\epsilon_a$ is a member of $S'$ that precedes $\epsilon_{i_j}$.
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Hence $\epsilon_{i_j} = \mathcal{E}_{\neg}(\epsilon_{i_a})$ for some
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$a < j$.
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\paragraph{Case 3}%
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Suppose $\epsilon_k = \mathcal{E}_{\square}(\epsilon_a, \epsilon_b)$ for
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some $a, b < k$ where $\square$ is one of the binary connectives
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$\land$, $\lor$, $\Rightarrow$, $\Leftrightarrow$.
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It must be that $A_4$ is found in neither $\epsilon_a$ nor $\epsilon_b$,
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else an immediate contradiction is raised.
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Therefore $\epsilon_a$ and $\epsilon_b$ is a member of $S'$, both of which
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precede $\epsilon_{i_j}$.
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Hence
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$\epsilon_{i_j} = \mathcal{E}_{\square}(\epsilon_{i_a}, \epsilon_{i_b})$
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for some $a, b < j$.
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\paragraph{Conclusion}%
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Since the above cases are exhaustive and apply to an arbitrary member of
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$S'$, it must be that every member of $S'$ is valid.
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Hence $S'$ is still a legal construction sequence.
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\end{proof}
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\subsection{\sorry{Exercise 1.1.5}}%
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\hyperlabel{sub:exercise-1.1.5}
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Suppose that $\alpha$ is a wff not containing the negation symbol $\neg$.
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\begin{enumerate}[(a)]
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\item Show that the length of $\alpha$ (i.e., the number of symbols in the
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string) is odd.
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\item Show that more than a quarter of the symbols are sentence symbols.
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\end{enumerate}
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\subsubsection{\sorry{Exercise 1.1.5a}}%
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\hyperlabel{ssub:exercise-1.1.5-a}
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Show that the length of $\alpha$ (i.e., the number of symbols in the string)
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is odd.
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\textit{Suggestion}: Apply induction to show that the length is of the form
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$4k + 1$ and the number of sentence symbols is $k + 1$.
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$4k + 1$.
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\begin{proof}
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TODO
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\end{proof}
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\subsubsection{\sorry{Exercise 1.1.6a}}%
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\hyperlabel{ssub:exercise-1.1.6-a}
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Show that more than a quarter of the symbols are sentence symbols.
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\textit{Suggestion}: Apply induction to show that the number of sentence
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symbols is $k + 1$.
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\begin{proof}
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TODO
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@ -1,10 +0,0 @@
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/-! # Enderton.Logic.Chapter_0
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Useful Facts About Sets
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-/
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namespace Enderton.Logic.Chapter_0
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-- TODO
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end Enderton.Logic.Chapter_0
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@ -0,0 +1,184 @@
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import Common.Nat.Basic
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import Mathlib.Data.Nat.Basic
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import Mathlib.Tactic.NormNum
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/-! # Enderton.Logic.Chapter_1
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Sentential Logic
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-/
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namespace Enderton.Logic.Chapter_1
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/--
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An abstract representation of a well-formed formula as defined by Enderton.
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-/
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inductive Wff where
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| SS : Nat → Wff -- e.g. **S**entence **S**ymbol `Aₙ`
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| Not : Wff → Wff -- e.g. `(¬ α)`
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| And : Wff → Wff → Wff -- e.g. `(α ∧ β)`
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| Or : Wff → Wff → Wff -- e.g. `(α ∨ β)`
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| Cond : Wff → Wff → Wff -- e.g. `(α → β)`
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| Iff : Wff → Wff → Wff -- e.g. `(α ↔ β)`
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namespace Wff
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/--
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Returns the length of the expression, including symbols.
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-/
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def length : Wff → ℕ
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| Wff.SS _ => 1
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| Wff.Not e => length e + 3
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| Wff.And e₁ e₂
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| Wff.Or e₁ e₂
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| Wff.Cond e₁ e₂
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| Wff.Iff e₁ e₂ => length e₁ + length e₂ + 3
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/--
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Every well-formed formula has a positive length.
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-/
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theorem length_gt_zero (φ : Wff)
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: length φ > 0 := by
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unfold length
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match φ with
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| SS _
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| Not _
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| And _ _
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| Or _ _
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| Cond _ _
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| Iff _ _ => simp
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end Wff
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/-! #### Exercise 1.1.2
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Show that there are no wffs of length `2`, `3`, or `6`, but that any other
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positive length is possible.
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-/
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section Exercise_1_1_2
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private lemma eq_3_by_cases (m n : ℕ) (h : m + n = 3)
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: m = 0 ∧ n = 3 ∨
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m = 1 ∧ n = 2 ∨
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m = 2 ∧ n = 1 ∨
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m = 3 ∧ n = 0 := by
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have m_le_3 : m ≤ 3 := by
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have : m = 3 - n := Eq.symm $ Nat.sub_eq_of_eq_add (Eq.symm h)
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rw [this]
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norm_num
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apply Or.elim (Nat.lt_or_eq_of_le m_le_3)
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· intro hm₁
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apply Or.elim (Nat.lt_or_eq_of_lt hm₁)
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· intro hm₂
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apply Or.elim (Nat.lt_or_eq_of_lt hm₂)
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· intro hm₃
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refine Or.elim (Nat.lt_or_eq_of_lt hm₃) (by simp) ?_
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intro m_eq_0
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rw [m_eq_0, zero_add] at h
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left
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exact ⟨m_eq_0, h⟩
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· intro m_eq_1
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rw [m_eq_1, add_comm] at h
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norm_num at h
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right; left
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exact ⟨m_eq_1, h⟩
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· intro m_eq_2
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rw [m_eq_2, add_comm] at h
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norm_num at h
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right; right; left
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exact ⟨m_eq_2, h⟩
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· intro m_eq_3
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rw [m_eq_3, add_comm] at h
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norm_num at h
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right; right; right
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exact ⟨m_eq_3, h⟩
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theorem exercise_1_1_2_i (φ : Wff)
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: φ.length ≠ 2 ∧ φ.length ≠ 3 ∧ φ.length ≠ 6 := by
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induction φ with
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| SS c =>
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unfold Wff.length
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simp
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| Not e ih =>
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unfold Wff.length
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refine ⟨by norm_num, ?_, ?_⟩
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· intro h
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norm_num at h
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have := e.length_gt_zero
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rw [h] at this
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simp at this
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· intro h
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norm_num at h
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rw [h] at ih
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simp at ih
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| And e₁ e₂ ih₁ ih₂
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| Or e₁ e₂ ih₁ ih₂
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| Cond e₁ e₂ ih₁ ih₂
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| Iff e₁ e₂ ih₁ ih₂ =>
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unfold Wff.length
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refine ⟨by norm_num, ?_, ?_⟩
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· intro h
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norm_num at h
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have := e₁.length_gt_zero
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rw [h.left] at this
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simp at this
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· intro h
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norm_num at h
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apply Or.elim (eq_3_by_cases e₁.length e₂.length h)
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· intro h₁
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have := e₁.length_gt_zero
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rw [h₁.left] at this
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simp at this
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· intro h₁
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apply Or.elim h₁
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· intro h₂
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exact absurd h₂.right ih₂.left
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intro h₂
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apply Or.elim h₂
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· intro h₃
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exact absurd h₃.left ih₁.left
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intro h₃
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exact absurd h₃.left ih₁.right.left
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theorem exercise_1_1_2_ii (n : ℕ) (h : n ≠ 2 ∧ n ≠ 3 ∧ n ≠ 6)
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: ∃ φ : Wff, φ.length = n := by
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let φ₁ := Wff.SS 1
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let φ₂ := Wff.And φ₁ (Wff.SS 2)
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let φ₃ := Wff.And φ₂ (Wff.SS 3)
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sorry
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end Exercise_1_1_2
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/-! #### Exercise 1.1.3
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Let `α` be a wff; let `c` be the number of places at which binary connective
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symbols (`∧`, `∨`, `→`, `↔`) occur in `α`; let `s` be the number of places at
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which sentence symbols occur in `α`. (For example, if `α` is `(A → (¬ A))` then
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`c = 1` and `s = 2`.) Show by using the induction principle that `s = c + 1`.
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-/
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section Exercise_1_1_3
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private def binary_symbol_count : Wff → ℕ
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| Wff.SS _ => 0
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| Wff.Not e => binary_symbol_count e
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| Wff.And e₁ e₂
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| Wff.Or e₁ e₂
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| Wff.Cond e₁ e₂
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| Wff.Iff e₁ e₂ => binary_symbol_count e₁ + binary_symbol_count e₂ + 1
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private def sentence_symbol_count : Wff → ℕ
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| Wff.SS _ => 1
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| Wff.Not e => sentence_symbol_count e
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| Wff.And e₁ e₂
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| Wff.Or e₁ e₂
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| Wff.Cond e₁ e₂
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| Wff.Iff e₁ e₂ => sentence_symbol_count e₁ + sentence_symbol_count e₂
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theorem exercise_1_1_3 (φ : Wff)
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: sentence_symbol_count φ = binary_symbol_count φ + 1 := by
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sorry
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end Exercise_1_1_3
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end Enderton.Logic.Chapter_1
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\textbf{Addition} ($+$) is the \nameref{ref:binary-operation} on $\omega$ such
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that for any $m$ and $n$ in $\omega$, $$m + n = A_m(n).$$
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\lean{Init/Prelude}{Add.add}
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\lean{Init/Prelude}
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{Add.add}
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\section{\defined{Axiom of Choice, First Form}}%
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\hyperlabel{ref:axiom-of-choice-1}
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@ -43,7 +44,8 @@
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For any relation $R$ there is a function $H \subseteq R$ with
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$\dom{H} = \dom{R}$.
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\lean*{Init/Prelude}{Classical.choice}
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\lean*{Init/Prelude}
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{Classical.choice}
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\section{\defined{Axiom of Choice, Second Form}}%
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\hyperlabel{ref:axiom-of-choice-2}
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@ -51,7 +53,8 @@
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For any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$
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for all $i \in I$, then $$\bigtimes_{i \in I} H(i) \neq \emptyset.$$
|
||||
|
||||
\lean{Init/Prelude}{Classical.choice}
|
||||
\lean{Init/Prelude}
|
||||
{Classical.choice}
|
||||
|
||||
\section{\defined{Binary Operation}}%
|
||||
\hyperlabel{ref:binary-operation}
|
||||
|
|
@ -79,7 +82,8 @@
|
|||
only if for all $x$ and $y$ in $A$,
|
||||
$$xRy \Rightarrow F(x)RF(y).$$
|
||||
|
||||
\lean{Init/Core}{Quotient.lift}
|
||||
\lean{Init/Core}
|
||||
{Quotient.lift}
|
||||
|
||||
\section{\defined{Composition}}%
|
||||
\hyperlabel{ref:composition}
|
||||
|
|
@ -87,9 +91,11 @@
|
|||
The \textbf{composition} of sets $F$ and $G$ is
|
||||
$$F \circ G = \{\tuple{u, v} \mid \exists t(uGt \land tFv)\}.$$
|
||||
|
||||
\code{Bookshelf/Enderton/Set/Relation}{Set.Relation.comp}
|
||||
\code{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.comp}
|
||||
|
||||
\lean{Mathlib/Data/Rel}{Rel.comp}
|
||||
\lean{Mathlib/Data/Rel}
|
||||
{Rel.comp}
|
||||
|
||||
\section{\defined{Connected}}%
|
||||
\hyperlabel{ref:connected}
|
||||
|
|
@ -105,9 +111,11 @@
|
|||
The \textbf{domain} of set $R$, denoted $\dom{R}$, is given by
|
||||
$$x \in \dom{R} \iff \exists y \tuple{x, y} \in R.$$
|
||||
|
||||
\code{Bookshelf/Enderton/Set/Relation}{Set.Relation.dom}
|
||||
\code{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.dom}
|
||||
|
||||
\lean{Mathlib/Data/Rel}{Rel.dom}
|
||||
\lean{Mathlib/Data/Rel}
|
||||
{Rel.dom}
|
||||
|
||||
\section{\defined{Empty Set Axiom}}%
|
||||
\hyperlabel{ref:empty-set-axiom}
|
||||
|
|
@ -122,7 +130,8 @@
|
|||
A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
||||
only if there is a one-to-one \nameref{ref:function} from $A$ onto $B$.
|
||||
|
||||
\lean*{Mathlib/Init/Function}{Function.Bijective}
|
||||
\lean*{Mathlib/Init/Function}
|
||||
{Function.Bijective}
|
||||
|
||||
\section{\defined{Equivalence Class}}%
|
||||
\hyperlabel{ref:equivalence-class}
|
||||
|
|
@ -133,7 +142,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
(\textbf{modulo $R$}).
|
||||
If the relation $R$ is fixed by the context, we may write just $[x]$.
|
||||
|
||||
\code*{Bookshelf/Enderton/Set/Relation}{Set.Relation.modEquiv}
|
||||
\code*{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.modEquiv}
|
||||
|
||||
\section{\defined{Equivalence Relation}}%
|
||||
\hyperlabel{ref:equivalence-relation}
|
||||
|
|
@ -142,7 +152,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
$R$ is a binary \nameref{ref:relation} on $A$ that is \nameref{ref:reflexive}
|
||||
on $A$, \nameref{ref:symmetric}, and \nameref{ref:transitive}.
|
||||
|
||||
\code*{Bookshelf/Enderton/Set/Relation}{Set.Relation.isEquivalence}
|
||||
\code*{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.isEquivalence}
|
||||
|
||||
\section{\defined{Exponentiation}}%
|
||||
\hyperlabel{ref:exponentiation}
|
||||
|
|
@ -157,7 +168,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
\textbf{Exponentiation} is the \nameref{ref:binary-operation} on $\omega$
|
||||
such that for any $m$ and $n$ in $\omega$, $$m^n = E_m(n).$$
|
||||
|
||||
\lean{Init/Prelude}{Pow.pow}
|
||||
\lean{Init/Prelude}
|
||||
{Pow.pow}
|
||||
|
||||
\section{\defined{Extensionality Axiom}}%
|
||||
\hyperlabel{ref:extensionality-axiom}
|
||||
|
|
@ -166,7 +178,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
$$\forall A, \forall B,
|
||||
\left[\forall x, (x \in A \iff x \in B) \Rightarrow A = B\right].$$
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.ext}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.ext}
|
||||
|
||||
\section{\defined{Field}}%
|
||||
\hyperlabel{ref:field}
|
||||
|
|
@ -174,7 +187,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
Given \nameref{ref:relation} $R$, the \textbf{field} of $R$, denoted $\fld{R}$,
|
||||
is given by $$\fld{R} = \dom{R} \cup \ran{R}.$$
|
||||
|
||||
\lean{Bookshelf/Enderton/Set/Relation}{Set.Relation.fld}
|
||||
\lean{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.fld}
|
||||
|
||||
\section{\defined{Finite Set}}%
|
||||
\hyperlabel{ref:finite-set}
|
||||
|
|
@ -182,7 +196,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
A set is \textbf{finite} if and only if it is \nameref{ref:equinumerous} to a
|
||||
\nameref{ref:natural-number}.
|
||||
|
||||
\lean{Mathlib/Data/Finset/Basic}{Finset}
|
||||
\lean{Mathlib/Data/Finset/Basic}
|
||||
{Finset}
|
||||
|
||||
\section{\defined{Function}}%
|
||||
\hyperlabel{ref:function}
|
||||
|
|
@ -200,11 +215,14 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
only one $x$ such that $xFy$.
|
||||
One-to-one functions are sometimes called \textbf{injections}.
|
||||
|
||||
\lean*{Mathlib/Init/Function}{Function.Injective}
|
||||
\lean*{Mathlib/Init/Function}
|
||||
{Function.Injective}
|
||||
|
||||
\lean{Mathlib/Init/Function}{Function.Surjective}
|
||||
\lean{Mathlib/Init/Function}
|
||||
{Function.Surjective}
|
||||
|
||||
\lean{Mathlib/Init/Function}{Function.Bijective}
|
||||
\lean{Mathlib/Init/Function}
|
||||
{Function.Bijective}
|
||||
|
||||
\section{\defined{Image}}%
|
||||
\hyperlabel{ref:image}
|
||||
|
|
@ -217,9 +235,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
& = \{v \mid (\exists u \in A) uFv\}.
|
||||
\end{align*}
|
||||
|
||||
\code{Bookshelf/Enderton/Set/Relation}{Set.Relation.image}
|
||||
\code{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.image}
|
||||
|
||||
\lean{Mathlib/Data/Rel}{Rel.image}
|
||||
\lean{Mathlib/Data/Rel}
|
||||
{Rel.image}
|
||||
|
||||
\section{\defined{Inductive Set}}%
|
||||
\hyperlabel{ref:inductive-set}
|
||||
|
|
@ -228,9 +248,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
and it is "closed under \nameref{ref:successor}", i.e.
|
||||
$$(\forall a \in A) a^+ \in A.$$
|
||||
|
||||
\lean{Prelude}{Nat}
|
||||
\lean{Prelude}
|
||||
{Nat}
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.univ}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.univ}
|
||||
|
||||
\begin{note}
|
||||
Induction is baked into Lean's type system.
|
||||
|
|
@ -252,9 +274,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
$$(\exists A)\left[
|
||||
\emptyset \in A \land (\forall a \in A) a^+ \in A\right].$$
|
||||
|
||||
\lean{Prelude}{Nat}
|
||||
\lean{Prelude}
|
||||
{Nat}
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.univ}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.univ}
|
||||
|
||||
\section{\defined{Inverse}}%
|
||||
\hyperlabel{ref:inverse}
|
||||
|
|
@ -262,9 +286,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
The \textbf{inverse} of a set $F$ is the set
|
||||
$$F^{-1} = \{\tuple{u, v} \mid vFu\}.$$
|
||||
|
||||
\code{Bookshelf/Enderton/Set/Relation}{Set.Relation.inv}
|
||||
\code{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.inv}
|
||||
|
||||
\lean{Mathlib/Data/Rel}{Rel.inv}
|
||||
\lean{Mathlib/Data/Rel}
|
||||
{Rel.inv}
|
||||
|
||||
\section{\defined{Irreflexive}}%
|
||||
\hyperlabel{ref:irreflexive}
|
||||
|
|
@ -272,7 +298,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
A binary relation $R$ on set $A$ is \textbf{irreflexive} if there is no
|
||||
$x \in A$ for which $xRx$.
|
||||
|
||||
\lean*{Mathlib/Init/Algebra/Classes}{IsIrrefl}
|
||||
\lean*{Mathlib/Init/Algebra/Classes}
|
||||
{IsIrrefl}
|
||||
|
||||
\section{\defined{Linear Ordering}}
|
||||
\hyperlabel{ref:linear-ordering}
|
||||
|
|
@ -287,7 +314,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
\item $R$ is \nameref{ref:trichotomous}.
|
||||
\end{enumerate}
|
||||
|
||||
\lean{Mathlib/Init/Algebra/Classes}{IsStrictTotalOrder}
|
||||
\lean{Mathlib/Init/Algebra/Classes}
|
||||
{IsStrictTotalOrder}
|
||||
|
||||
\begin{note}
|
||||
This definition does not agree with how Lean defines a linear order.
|
||||
|
|
@ -320,7 +348,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
\textbf{Multiplication} ($\cdot$) is the \nameref{ref:binary-operation} on
|
||||
$\omega$ such that for any $m$ and $n$ in $\omega$, $$m \cdot n = M_m(n).$$
|
||||
|
||||
\lean{Init/Prelude}{Mul.mul}
|
||||
\lean{Init/Prelude}
|
||||
{Mul.mul}
|
||||
|
||||
\section{\defined{Natural Number}}%
|
||||
\hyperlabel{ref:natural-number}
|
||||
|
|
@ -329,7 +358,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
The set of all natural numbers exists by virtue of \nameref{sub:theorem-4a}.
|
||||
This set is denoted as $\omega$.
|
||||
|
||||
\lean*{Prelude}{Nat}
|
||||
\lean*{Prelude}
|
||||
{Nat}
|
||||
|
||||
\section{\defined{Ordered Pair}}%
|
||||
\hyperlabel{ref:ordered-pair}
|
||||
|
|
@ -337,9 +367,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
For any sets $u$ and $v$, the \textbf{ordered pair} $\tuple{u, v}$ is
|
||||
the set $\{\{u\}, \{u, v\}\}$.
|
||||
|
||||
\code*{Bookshelf/Enderton/Set/OrderedPair}{OrderedPair}
|
||||
\code*{Bookshelf/Enderton/Set/OrderedPair}
|
||||
{OrderedPair}
|
||||
|
||||
\lean{Prelude}{Prod}
|
||||
\lean{Prelude}
|
||||
{Prod}
|
||||
|
||||
\section{\defined{Ordering on \texorpdfstring{$\omega$}{Natural Numbers}}}%
|
||||
\hyperlabel{ref:ordering-natural-numbers}
|
||||
|
|
@ -356,7 +388,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
& \iff m < n \lor m = n.
|
||||
\end{align*}
|
||||
|
||||
\lean{Init/Prelude}{Nat.lt}
|
||||
\lean{Init/Prelude}
|
||||
{Nat.lt}
|
||||
|
||||
\section{\defined{Pair Set}}%
|
||||
\hyperlabel{ref:pair-set}
|
||||
|
|
@ -364,9 +397,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
For any sets $u$ and $v$, the \textbf{pair set $\{u, v\}$} is the set whose
|
||||
only members are $u$ and $v$.
|
||||
|
||||
\lean*{Mathlib/Init/Set}{Set.insert}
|
||||
\lean*{Mathlib/Init/Set}
|
||||
{Set.insert}
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.singleton}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.singleton}
|
||||
|
||||
\section{\defined{Pairing Axiom}}%
|
||||
\hyperlabel{ref:pairing-axiom}
|
||||
|
|
@ -375,9 +410,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
$$\forall u, \forall v, \exists B, \forall x,
|
||||
(x \in B \iff x = u \text{ or } x = v).$$
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.insert}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.insert}
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.singleton}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.singleton}
|
||||
|
||||
\section{\defined{Partition}}%
|
||||
\hyperlabel{ref:partition}
|
||||
|
|
@ -389,7 +426,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
\item each element of $A$ is in some set in $\Pi$.
|
||||
\end{enumerate}
|
||||
|
||||
\lean{Mathlib/Data/Setoid/Partition}{Setoid.IsPartition}
|
||||
\code{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.Partition}
|
||||
|
||||
\lean{Mathlib/Data/Setoid/Partition}
|
||||
{Setoid.IsPartition}
|
||||
|
||||
\section{\defined{Peano System}}%
|
||||
\hyperlabel{ref:peano-system}
|
||||
|
|
@ -404,7 +445,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
itself.
|
||||
\end{enumerate}
|
||||
|
||||
\code{Common/Set/Peano}{Peano.System}
|
||||
\code{Common/Set/Peano}
|
||||
{Peano.System}
|
||||
|
||||
\section{\defined{Power Set}}%
|
||||
\hyperlabel{ref:power-set}
|
||||
|
|
@ -412,7 +454,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
For any set $a$, the \textbf{power set $\powerset{a}$} is the set whose
|
||||
members are exactly the subsets of $a$.
|
||||
|
||||
\lean*{Mathlib/Init/Set}{Set.powerset}
|
||||
\lean*{Mathlib/Init/Set}
|
||||
{Set.powerset}
|
||||
|
||||
\section{\defined{Power Set Axiom}}%
|
||||
\hyperlabel{ref:power-set-axiom}
|
||||
|
|
@ -420,7 +463,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
For any set $a$, there is a set whose members are exactly the subsets of $a$:
|
||||
$$\forall a, \exists B, \forall x, (x \in B \iff x \subseteq a).$$
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.powerset}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.powerset}
|
||||
|
||||
\section{\defined{Proper Subset}}%
|
||||
\hyperlabel{ref:proper-subset}
|
||||
|
|
@ -429,7 +473,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
only if it is a subset of $B$ that is unequal to $B$.
|
||||
$$A \subset B \iff A \subseteq B \land A \neq B.$$
|
||||
|
||||
\lean{Std/Classes/SetNotation}{HasSSubset}
|
||||
\lean{Std/Classes/SetNotation}
|
||||
{HasSSubset}
|
||||
|
||||
\section{\defined{Quotient Set}}%
|
||||
\hyperlabel{ref:quotient-set}
|
||||
|
|
@ -439,7 +484,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
are the equivalence classes.
|
||||
The expression $A / R$ is read "$A$ modulo $R$.
|
||||
|
||||
\lean*{Init/Core}{Quotient}
|
||||
\lean*{Init/Core}
|
||||
{Quotient}
|
||||
|
||||
\section{\defined{Range}}%
|
||||
\hyperlabel{ref:range}
|
||||
|
|
@ -447,9 +493,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
The \textbf{range} of set $R$, denoted $\ran{R}$, is given by
|
||||
$$x \in \ran{R} \iff \exists t \tuple{t, x} \in R.$$
|
||||
|
||||
\code{Bookshelf/Enderton/Set/Relation}{Set.Relation.ran}
|
||||
\code{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.ran}
|
||||
|
||||
\lean{Mathlib/Data/Rel}{Rel.codom}
|
||||
\lean{Mathlib/Data/Rel}
|
||||
{Rel.codom}
|
||||
|
||||
\section{\defined{Reflexive}}%
|
||||
\hyperlabel{ref:reflexive}
|
||||
|
|
@ -457,16 +505,19 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
A binary relation $R$ is \textbf{reflexive} on $A$ if and only if $xRx$ for
|
||||
all $x \in A$.
|
||||
|
||||
\code*{Bookshelf/Enderton/Set/Relation}{Set.Relation.isReflexive}
|
||||
\code*{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.isReflexive}
|
||||
|
||||
\lean{Mathlib/Init/Algebra/Classes}{IsRefl}
|
||||
\lean{Mathlib/Init/Algebra/Classes}
|
||||
{IsRefl}
|
||||
|
||||
\section{\defined{Relation}}%
|
||||
\hyperlabel{ref:relation}
|
||||
|
||||
A \textbf{relation} is a set of \nameref{ref:ordered-pair}s.
|
||||
|
||||
\code*{Bookshelf/Enderton/Set/Relation}{Set.Relation}
|
||||
\code*{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation}
|
||||
|
||||
\lean{Mathlib/Data/Rel}{Rel}
|
||||
|
||||
|
|
@ -476,14 +527,16 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
The \textbf{restriction} of a set $F$ to set $A$ is the set
|
||||
$$F \restriction A = \{\tuple{u, v} \mid uFv \land u \in A\}.$$
|
||||
|
||||
\code{Bookshelf/Enderton/Set/Relation}{Set.Relation.restriction}
|
||||
\code{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.restriction}
|
||||
|
||||
\section{\defined{Successor}}%
|
||||
\hyperlabel{ref:successor}
|
||||
|
||||
For any set $a$, its \textbf{successor} is defined by $$a^+ = a \cup \{a\}.$$
|
||||
|
||||
\lean{Prelude}{Nat.succ}
|
||||
\lean{Prelude}
|
||||
{Nat.succ}
|
||||
|
||||
\begin{note}
|
||||
The corresponding Lean definition refers to the `Nat.succ` constructor.
|
||||
|
|
@ -498,7 +551,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
$$\forall t_1, \cdots \forall t_k, \forall c,
|
||||
\exists B, \forall x, (x \in B \iff x \in c \land \phi).$$
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.Subset}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.Subset}
|
||||
|
||||
\section{\defined{Symmetric}}%
|
||||
\hyperlabel{ref:symmetric}
|
||||
|
|
@ -506,7 +560,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
A binary relation $R$ is \textbf{symmetric} if and only if whenever $xRy$ then
|
||||
$yRx$.
|
||||
|
||||
\code*{Bookshelf/Enderton/Set/Relation}{Set.Relation.isSymmetric}
|
||||
\code*{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.isSymmetric}
|
||||
|
||||
\section{\defined{Symmetric Difference}}%
|
||||
\hyperlabel{ref:symmetric-difference}
|
||||
|
|
@ -514,7 +569,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
The \textbf{symmetric difference} $A + B$ of sets $A$ and $B$ is the set
|
||||
$(A - B) \cup (B - A)$.
|
||||
|
||||
\lean*{Mathlib/Data/Set/Basic}{symmDiff\_def}
|
||||
\lean*{Mathlib/Data/Set/Basic}
|
||||
{symmDiff\_def}
|
||||
|
||||
\section{\defined{Transitive}}%
|
||||
\hyperlabel{ref:transitive}
|
||||
|
|
@ -522,9 +578,11 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
A binary relation $R$ is \textbf{transitive} if and only if whenever $xRy$ and
|
||||
$yRz$, then $xRz$.
|
||||
|
||||
\code*{Bookshelf/Enderton/Set/Relation}{Set.Relation.isTransitive}
|
||||
\code*{Bookshelf/Enderton/Set/Relation}
|
||||
{Set.Relation.isTransitive}
|
||||
|
||||
\lean{Mathlib/Init/Algebra/Classes}{IsTrans}
|
||||
\lean{Mathlib/Init/Algebra/Classes}
|
||||
{IsTrans}
|
||||
|
||||
\section{\defined{Transitive Set}}%
|
||||
\hyperlabel{ref:transitive-set}
|
||||
|
|
@ -541,7 +599,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
$$xRy, \quad x = y, \quad yRx$$
|
||||
holds.
|
||||
|
||||
\lean*{Mathlib/Init/Algebra/Classes}{IsTrichotomous}
|
||||
\lean*{Mathlib/Init/Algebra/Classes}
|
||||
{IsTrichotomous}
|
||||
|
||||
\section{\defined{Union Axiom}}%
|
||||
\hyperlabel{ref:union-axiom}
|
||||
|
|
@ -551,7 +610,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
$$\forall A, \exists B, \forall x
|
||||
\left[ x \in B \iff (\exists b \in A) x \in b \right]$$
|
||||
|
||||
\lean{Mathlib/Data/Set/Lattice}{Set.sUnion}
|
||||
\lean{Mathlib/Data/Set/Lattice}
|
||||
{Set.sUnion}
|
||||
|
||||
\section{\defined{Union Axiom, Preliminary Form}}%
|
||||
\hyperlabel{ref:union-axiom-preliminary-form}
|
||||
|
|
@ -561,7 +621,8 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
|
|||
$$\forall a, \forall b, \exists B, \forall x,
|
||||
(x \in B \iff x \in a \text{ or } x \in b).$$
|
||||
|
||||
\lean{Mathlib/Init/Set}{Set.union}
|
||||
\lean{Mathlib/Init/Set}
|
||||
{Set.union}
|
||||
|
||||
\endgroup
|
||||
|
||||
|
|
|
|||
|
|
@ -1941,7 +1941,7 @@ Show that `Rₚ` is an equivalence relation on `A`. (This is a formalized versio
|
|||
of the discussion at the beginning of this section.)
|
||||
-/
|
||||
theorem exercise_3_37 {P : Set (Set α)} {A : Set α}
|
||||
(hP : isPartition P A) (Rₚ : Set.Relation α)
|
||||
(hP : Partition P A) (Rₚ : Set.Relation α)
|
||||
(hRₚ : ∀ x y, (x, y) ∈ Rₚ ↔ ∃ B ∈ P, x ∈ B ∧ y ∈ B)
|
||||
: isEquivalence Rₚ A := by
|
||||
have hRₚ_eq : Rₚ = { p | ∃ B ∈ P, p.1 ∈ B ∧ p.2 ∈ B } := by
|
||||
|
|
@ -2014,7 +2014,7 @@ equivalence relation on `A`. Show that if we start with the equivalence relation
|
|||
`Rₚ` of the preceding exercise, then the partition `A / Rₚ` is just `P`.
|
||||
-/
|
||||
theorem exercise_3_38 {P : Set (Set α)} {A : Set α}
|
||||
(hP : isPartition P A) (Rₚ : Set.Relation α)
|
||||
(hP : Partition P A) (Rₚ : Set.Relation α)
|
||||
(hRₚ : ∀ x y, (x, y) ∈ Rₚ ↔ ∃ B ∈ P, x ∈ B ∧ y ∈ B)
|
||||
: modEquiv (exercise_3_37 hP Rₚ hRₚ) = P := by
|
||||
have hRₚ_eq : Rₚ = { p | ∃ B ∈ P, p.1 ∈ B ∧ p.2 ∈ B } := by
|
||||
|
|
|
|||
|
|
@ -575,7 +575,7 @@ theorem neighborhood_mem_imp_relate {R : Set.Relation α} {A : Set α}
|
|||
A **partition** `Π` of a set `A` is a set of nonempty subsets of `A` that is
|
||||
disjoint and exhaustive.
|
||||
-/
|
||||
structure isPartition (P : Set (Set α)) (A : Set α) : Prop where
|
||||
structure Partition (P : Set (Set α)) (A : Set α) : Prop where
|
||||
p_subset : ∀ p ∈ P, p ⊆ A
|
||||
nonempty : ∀ p ∈ P, Set.Nonempty p
|
||||
disjoint : ∀ a ∈ P, ∀ b, b ∈ P → a ≠ b → a ∩ b = ∅
|
||||
|
|
@ -585,7 +585,7 @@ structure isPartition (P : Set (Set α)) (A : Set α) : Prop where
|
|||
Membership of sets within `P` is unique.
|
||||
-/
|
||||
theorem partition_mem_mem_eq {P : Set (Set α)} {A : Set α}
|
||||
(hP : isPartition P A) (hx : x ∈ A)
|
||||
(hP : Partition P A) (hx : x ∈ A)
|
||||
: ∃! B, B ∈ P ∧ x ∈ B := by
|
||||
have ⟨B, hB⟩ := hP.exhaustive x hx
|
||||
refine ⟨B, hB, ?_⟩
|
||||
|
|
@ -606,7 +606,7 @@ def modEquiv {A : Set α} {R : Relation α} (_ : isEquivalence R A) :=
|
|||
Show the sets formed by `modEquiv` do indeed form a `partition`.
|
||||
-/
|
||||
theorem modEquiv_partition {A : Set α} {R : Relation α} (hR : isEquivalence R A)
|
||||
: isPartition (modEquiv hR) A := by
|
||||
: Partition (modEquiv hR) A := by
|
||||
refine ⟨?_, ?_, ?_, ?_⟩
|
||||
· intro p hp
|
||||
have ⟨x, hx⟩ := hp
|
||||
|
|
|
|||
|
|
@ -1,7 +1,14 @@
|
|||
import Mathlib.Data.Set.Basic
|
||||
import Mathlib.Tactic.NormNum
|
||||
|
||||
namespace Nat
|
||||
|
||||
/--
|
||||
If `n < m⁺`, then `n < m` or `n = m`.
|
||||
-/
|
||||
theorem lt_or_eq_of_lt {n m : Nat} (h : n < m.succ) : n < m ∨ n = m :=
|
||||
lt_or_eq_of_le (lt_succ.mp h)
|
||||
|
||||
/--
|
||||
The following cancellation law holds for `m`, `n`, and `p` in `ω`:
|
||||
```
|
||||
|
|
|
|||
|
|
@ -44,6 +44,7 @@ def index : BaseHtmlM Html := do templateExtends (baseHtml "Index") <|
|
|||
<li>Smullyan, Raymond M. To Mock a Mockingbird: And Other Logic Puzzles Including an Amazing Adventure in Combinatory Logic. Oxford: Oxford university press, 2000.</li>
|
||||
</ul>
|
||||
|
||||
<h2>Legend</h2>
|
||||
<p>
|
||||
A color/symbol code is used on generated PDF headers to indicate their
|
||||
status:
|
||||
|
|
|
|||
Loading…
Reference in New Issue