Enderton. Corollary 4P.

finite-set-exercises
Joshua Potter 2023-08-08 08:56:13 -06:00
parent f23d925544
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4 changed files with 72 additions and 24 deletions

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@ -7623,26 +7623,22 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\subparagraph{($\Leftarrow$)}% \subparagraph{($\Leftarrow$)}%
Let $p$ be a natural number and suppose $m + p \in n + p$. Let $p$ be a natural number and suppose $m + p \in n + p$.
By the \nameref{sub:trichotomy-law-natural-numbers}, there are three By the \nameref{sub:trichotomy-law-natural-numbers}, there are two
ways $m$ and $n$ may relate to one another. cases to consider regarding how $m$ and $n$ relate to one another:
\vspace{8pt} \vspace{8pt}
\textbf{Case 1}: Suppose $m \in n$. \textbf{Case 1}: Suppose $m = n$.
Then our proof is complete.
\vspace{8pt}
\textbf{Case 2}: Suppose $m = n$.
Then $m + p \in n + p = m + p$. Then $m + p \in n + p = m + p$.
\nameref{sub:lemma-4l-b} shows this is impossible. \nameref{sub:lemma-4l-b} shows this is impossible.
\vspace{8pt} \vspace{8pt}
\textbf{Case 3}: Suppose $n \in m$. \textbf{Case 2}: Suppose $n \in m$.
Then \nameref{spar:theorem-4n-i-right} indicates $n + p \in m + p$. Then \nameref{spar:theorem-4n-i-right} indicates $n + p \in m + p$.
But this contradicts \nameref{sub:trichotomy-law-natural-numbers} since, But this contradicts \nameref{sub:trichotomy-law-natural-numbers} since,
by hypothesis, $m + p \in n + p$. by hypothesis, $m + p \in n + p$.
\vspace{8pt} \vspace{8pt}
\textbf{Conclusion}: The only possibility is $m \in n$. \textbf{Conclusion}: By trichotomy, it follows $m \in n$.
\paragraph{(ii)}% \paragraph{(ii)}%
\hyperlabel{par:theorem-4n-ii} \hyperlabel{par:theorem-4n-ii}
@ -7679,46 +7675,72 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\subparagraph{($\Leftarrow$)}% \subparagraph{($\Leftarrow$)}%
Let $p \neq 0$ be a natural number and suppose $m \cdot p \in n \cdot p$. Let $p \neq 0$ be a natural number and suppose $m \cdot p \in n \cdot p$.
By the \nameref{sub:trichotomy-law-natural-numbers}, there are three By the \nameref{sub:trichotomy-law-natural-numbers}, there are two
ways $m$ and $n$ may relate to one another. cases to consider regarding how $m$ and $n$ relate to one another:
\vspace{8pt} \vspace{8pt}
\textbf{Case 1}: Suppose $m \in n$. \textbf{Case 1}: Suppose $m = n$.
Then our proof is complete.
\vspace{8pt}
\textbf{Case 2}: Suppose $m = n$.
Then $m \cdot p \in n \cdot p = m \cdot p$. Then $m \cdot p \in n \cdot p = m \cdot p$.
\nameref{sub:lemma-4l-b} shows this is impossible. \nameref{sub:lemma-4l-b} shows this is impossible.
\vspace{8pt} \vspace{8pt}
\textbf{Case 3}: Suppose $n \in m$. \textbf{Case 2}: Suppose $n \in m$.
Then \nameref{spar:theorem-4n-ii-right} indicates Then \nameref{spar:theorem-4n-ii-right} indicates
$n \cdot p \in m \cdot p$. $n \cdot p \in m \cdot p$.
But this contradicts \nameref{sub:trichotomy-law-natural-numbers} since, But this contradicts \nameref{sub:trichotomy-law-natural-numbers} since,
by hypothesis, $m \cdot p \in n \cdot p$. by hypothesis, $m \cdot p \in n \cdot p$.
\vspace{8pt} \vspace{8pt}
\textbf{Conclusion}: The only possibility is $m \in n$. \textbf{Conclusion}: By trichotomy, it follows $m \in n$.
\end{proof} \end{proof}
\subsection{\sorry{Corollary 4P}}% \subsection{\verified{Corollary 4P}}%
\hyperlabel{sub:corollary-4p} \hyperlabel{sub:corollary-4p}
\begin{corollary}[4P] \begin{corollary}[4P]
The following cancellation laws hold for $m$, $n$, and $p$ in $\omega$: The following cancellation laws hold for $m$, $n$, and $p$ in $\omega$:
\begin{align*} \begin{align}
m + p = n + p & \Rightarrow m = n, \\ m + p = n + p & \Rightarrow m = n,
m \cdot p = n \cdot p & \Rightarrow m = n. & \hyperlabel{sub:corollary-4p-eq1} \\
\end{align*} m \cdot p = n \cdot p \land p \neq 0 & \Rightarrow m = n.
& \hyperlabel{sub:corollary-4p-eq2}
\end{align}
\end{corollary} \end{corollary}
\begin{proof} \begin{proof}
TODO \statementpadding
\lean*{Init/Data/Nat/Basic}{Nat.add\_right\_cancel}
\lean{Common/Nat/Basic}{Nat.mul\_right\_cancel}
\paragraph{\eqref{sub:corollary-4p-eq1}}%
Suppose $m + p = n + p$.
By the \nameref{sub:trichotomy-law-natural-numbers}, there are two
cases to consider regarding how $m$ and $n$ relate to one another.
If $m \in n$, then \nameref{sub:theorem-4n} implies $m + p \in n + p$.
If $n \in m$, then \nameref{sub:theorem-4n} implies $n + p \in m + p$.
Both of these contradict the \nameref{sub:trichotomy-law-natural-numbers} of
$m + p$ and $n + p$.
Thus $m = n$ is the only remaining possibility.
\paragraph{\eqref{sub:corollary-4p-eq2}}%
Suppose $m \cdot p = n \cdot p$ and $p \neq 0$.
By the \nameref{sub:trichotomy-law-natural-numbers}, there are two
cases to consider regarding how $m$ and $n$ relate to one another.
If $m \in n$, then \nameref{sub:theorem-4n} implies
$m \cdot p \in n \cdot p$.
If $n \in m$, then \nameref{sub:theorem-4n} implies
$n \cdot p \in m \cdot p$.
Both of these contradict the \nameref{sub:trichotomy-law-natural-numbers} of
$m \cdot p$ and $n \cdot p$.
Thus $m = n$ is the only remaining possibility.
\end{proof} \end{proof}

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import Common.List import Common.List
import Common.Logic import Common.Logic
import Common.Nat
import Common.Real import Common.Real
import Common.Set import Common.Set

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import Common.Nat.Basic

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import Mathlib.Data.Set.Basic
namespace Nat
/--
The following cancellation law holds for `m`, `n`, and `p` in `ω`:
```
m ⬝ p = n ⬝ p ∧ p ≠ 0 → m = n
```
-/
theorem mul_right_cancel (m n p : ) (hp : 0 < p) : m * p = n * p → m = n := by
intro hmn
match @trichotomous LT.lt _ m n with
| Or.inl h =>
have : m * p < n * p := Nat.mul_lt_mul_of_pos_right h hp
rw [hmn] at this
simp at this
| Or.inr (Or.inl h) => exact h
| Or.inr (Or.inr h) =>
have : n * p < m * p := Nat.mul_lt_mul_of_pos_right h hp
rw [hmn] at this
simp at this
end Nat