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typo #5

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36 changes: 18 additions & 18 deletions blueprint/src/rationals.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -36,7 +36,7 @@ \section{The \emph{prerationals}}
\begin{proof}
\leanok
\uses{MyPrerat.R}
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand All @@ -48,7 +48,7 @@ \section{The \emph{prerationals}}
\begin{proof}
\leanok
\uses{MyPrerat.R}
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand All @@ -60,7 +60,7 @@ \section{The \emph{prerationals}}
\begin{proof}
\leanok
\uses{MyPrerat.R}
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand Down Expand Up @@ -98,7 +98,7 @@ \section{The \emph{prerationals}}
\begin{proof}
\uses{MyPrerat.neg}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{definition}
Expand All @@ -107,7 +107,7 @@ \section{The \emph{prerationals}}
\leanok
\uses{MyPrerat.R}
We define an operation, called \emph{addition} on $\MyPrerat$ as follows: the addition of $x = (a,b)$
and $y = (b, c)$ is
and $y = (c, d)$ is
\[
x + y = (a,b) + (c,d) = (a * d + b * c, b*d)
\]
Expand All @@ -123,7 +123,7 @@ \section{The \emph{prerationals}}
\begin{proof}
\uses{MyPrerat.add}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{definition}
Expand All @@ -146,7 +146,7 @@ \section{The \emph{prerationals}}
\begin{proof}
\uses{MyPrerat.mul}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{definition}
Expand All @@ -170,7 +170,7 @@ \section{The \emph{prerationals}}
\begin{proof}
\uses{MyPrerat.inv}
\leanok
Exercice.
Exercise.
\end{proof}

\section{The rationals}
Expand Down Expand Up @@ -449,7 +449,7 @@ \subsection{\texorpdfstring{The inclusion $j \colon \MyInt \to \MyRat$}{The incl
\begin{proof}
\uses{MyRat.add, MyRat.j}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand All @@ -464,7 +464,7 @@ \subsection{\texorpdfstring{The inclusion $j \colon \MyInt \to \MyRat$}{The incl
\begin{proof}
\uses{MyRat.mul, MyRat.j}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand All @@ -476,7 +476,7 @@ \subsection{\texorpdfstring{The inclusion $j \colon \MyInt \to \MyRat$}{The incl
\begin{proof}
\uses{MyRat.j}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand All @@ -500,7 +500,7 @@ \subsection{\texorpdfstring{The inclusion $j \colon \MyInt \to \MyRat$}{The incl
\begin{proof}
\uses{MyRat.inv, MyRat.mul, MyRat.j}
\leanok
Exercice.
Exercise.
\end{proof}

\subsection{Nonegativity}
Expand Down Expand Up @@ -562,7 +562,7 @@ \subsection{Nonegativity}
\end{lemma}
\begin{proof}
\leanok
Annoying but easy, left as an exercice.
Annoying but easy, left as an exercise.
\end{proof}

\begin{lemma}
Expand All @@ -574,7 +574,7 @@ \subsection{Nonegativity}
\end{lemma}
\begin{proof}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand All @@ -586,7 +586,7 @@ \subsection{Nonegativity}
\end{lemma}
\begin{proof}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand All @@ -598,7 +598,7 @@ \subsection{Nonegativity}
\end{lemma}
\begin{proof}
\leanok
Exercice.
Exercise.
\end{proof}

\subsection{The order}
Expand Down Expand Up @@ -713,7 +713,7 @@ \subsection{Interaction between the order and the inclusions}
\begin{proof}
\uses{MyRat.IsNonneg, MyInt.le}
\leanok
Exercice.
Exercise.
\end{proof}

\begin{lemma}
Expand Down Expand Up @@ -763,7 +763,7 @@ \subsection{The linear order structure}
\begin{proof}
\leanok
\uses{MyRat.zero_le_iff_IsNonneg, MyRat.isNonneg_mul_isNonneg}
Exercice.
Exercise.
\end{proof}

We now have that $\MyRat$ is a \emph{strict ordered ring}: a nontrivial ring with a partial order such that addition is strictly monotone and multiplication by a positive number is strictly monotone.