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make those last changes compile

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commit 2b78bc4db2ee75c5cd1a499172820c1ea4cf6ac6 1 parent 138e543
@mikeshulman mikeshulman authored
Showing with 11 additions and 12 deletions.
  1. +11 −11 equivalences.tex
  2. +0 −1  univalence.tex
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22 equivalences.tex
@@ -122,7 +122,7 @@ \section{Adjoint Isomorphisms}
\begin{cor}\label{cor:equivs-equiv}
For types $A,B$ we have the following sequence of two equivalences and a logical equivalence
between our four equivalence relations.
- \[ (A\simeq B) \simeq (A\cong_a B) \simeq (A\cong_w B) \lra (A\cong B).\]
+ \[ (A\simeq B) \simeq (A\cong_a B) \simeq (A\cong_w B) \leftrightarrow (A\cong B).\]
\end{cor}
\begin{proof} ??
@@ -133,36 +133,36 @@ \section{Identity Systems on a Type Universe}
Let $\bbU$ be a type universe.
\begin{defn} An {\em identity system $(R,\sfr{})$ on $\bbU$} consists of a type $R_{A,B}$ for $A,B:\bbU$, together with $\sfr{A}:R_{A,A}$ for $A:\bbU$, such that the following holds.
-\be{quote}
+\begin{quote}
If $D_{A,B}(e)$ is a type for $A,B:\bbU$ and $e:R_{A,B}$ then there is
\[ J_{A,B}(e):D_{A,B}(e)\mbox{ for } A,B:\bbU \mbox{ and } e:R_{A,B},\]
such that
\[ J_{A,A}(\sfr{A})=d_A\mbox{ for } A:\bbU.\]
-\en{quote}
+\end{quote}
\end{defn}
\begin{eg}
$(Id_\bbU,\refl{\bbU})$ is an identity system on $\bbU$.
\end{eg}
-\be{thm}
+\begin{thm}
If $(R,\sfr{})$ is an identity system on $\bbU$ then
-\be{enumerate}
+\begin{enumerate}
\item $R_{A,B}\ra \eqv{A}{B}$ for $A,B:\bbU$
\item If $f_e:A\ra B$ and $g_e:B\ra A$ for $A,B:\bbU$ and $e:R_{A,B}$ such that
\[ f_{\sfr{A}} = g_{\sfr{A}}=1_A\mbox{ for } A:\bbU\]
then
\[ g_e\circ f_e =1_A\mbox{ and } f_e\circ g_e = 1_B
\mbox{ for } A,B:\bbU \mbox{ and } e:R_{A,B} \]
-\en{enumerate}
-\en{thm}
+\end{enumerate}
+\end{thm}
\newcommand{\sfequiv}[1]{{\sf reflequiv}^\bbU_{#1}}
-\be{rmk} Once we have the notion of a univalent type universe we can get the following result.
-\en{rmk}
-\be{thm}
+\begin{rmk} Once we have the notion of a univalent type universe we can get the following result.
+\end{rmk}
+\begin{thm}
$\bbU$ is a univalent type universe iff $(Equiv^\bbU,\sfequiv{})$ is an identity system on $\bbU$, where
\[ Equiv^\bbU_{A,B}\defeq (\eqv{A}{B})\mbox{ for } A,B:\bbU\]
and, if $s_A: (1_A\mbox{ is an equivalence }A\ra A)$,
\[ \sfequiv{A}\defeq (1_A,s_A):\eqv{A}{A}\mbox{ for } A:\bbU.\]
-\en{thm}
+\end{thm}
Using the two theorems I believe (from a still mysterious coq proof of Assia and Cyril) that we can get a quick proof of function extensionality on a univalent universe.
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1  univalence.tex
@@ -1,7 +1,6 @@
\chapter{Univalence}
\label{cha:univalence}
-\input{pa-macros}
\input{ua.tex}
\input{ua-fe.tex}
\input{ua-append}
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