409 lines
14 KiB
TeX
409 lines
14 KiB
TeX
\chapter{Cubical Agda}
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\section{Propositional equality}
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Judgmental equality in Agda is a feature of the type system. Its something that
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can be checked automatically by the type checker: In the example from the
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introduction $n + 0$ can be judged to be equal to $n$ simply by expanding the
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definition of $+$.
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On the other hand, propositional equality is something defined within the
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language itself. Propositional equality cannot be derived automatically. The
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normal definition of judgmental equality is an inductive data type. Cubical Agda
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discards this type in favor of a new primitives that has certain computational
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properties exclusive to it.
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Exceprts of the source code relevant to this section can be found in appendix
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\S\ref{sec:app-cubical}.
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\subsection{The equality type}
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The usual notion of judgmental equality says that given a type $A \tp \MCU$ and
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two points of $A$; $a_0, a_1 \tp A$ we can form the type:
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%
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\begin{align}
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a_0 \equiv a_1 \tp \MCU
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\end{align}
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%
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In Agda this is defined as an inductive data type with the single constructor
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for any $a \tp A$:
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%
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\begin{align}
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\refl \tp a \equiv a
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\end{align}
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%
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There also exist a related notion of \emph{heterogeneous} equality which allows
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for equating points of different types. In this case given two types $A, B \tp
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\MCU$ and two points $a \tp A$, $b \tp B$ we can construct the type:
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%
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\begin{align}
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a \cong b \tp \MCU
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\end{align}
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%
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This is likewise defined as an inductive data type with a single constructors
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for any $a \tp A$:
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%
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\begin{align}
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\refl \tp a \cong a
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\end{align}
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%
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In Cubical Agda these two notions are paralleled with homogeneous- and
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heterogeneous paths respectively.
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%
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\subsection{The path type}
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Judgmental equality in Cubical Agda is encapsulated with the type:
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%
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\begin{equation}
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\Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
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\end{equation}
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%
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$I$ is a special data type (\TODO{that also has special computational properties
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AFAIK}) called the index set. $I$ can be thought of simply as the interval on
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the real numbers from $0$ to $1$. $P$ is a family of types over the index set
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$I$. I will sometimes refer to $P$ as the \nomenindex{path space} of some path $p \tp
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\Path\ P\ a\ b$. By this token $P\ 0$ then corresponds to the type at the
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left-endpoint and $P\ 1$ as the type at the right-endpoint. The type is called
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$\Path$ because it is connected with paths in homotopy theory. The intuition
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behind this is that $\Path$ describes paths in $\MCU$ -- i.e.\ between types. For
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a path $p$ for the point $p\ i$ the index $i$ describes how far along the path
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one has moved. An inhabitant of $\Path\ P\ a_0\ a_1$ is a (dependent-) function,
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$p$, from the index-space to the path space:
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%
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$$
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p \tp \prod_{i \tp I} P\ i
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$$
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%
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Which must satisfy being judgmentally equal to $a_0$ (respectively $a_1$) at the
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endpoints. I.e.:
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%
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\begin{align*}
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p\ 0 & = a_0 \\
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p\ 1 & = a_1
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\end{align*}
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%
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The notion of \nomenindex{homogeneous equalities} is recovered when $P$ does not
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depend on its argument. That is for $A \tp \MCU$, $a_0, a_1 \tp A$ the
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homogenous equality between $a_0$ and $a_1$ is the type:
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%
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$$
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a_0 \equiv a_1 \defeq \Path\ (\lambda i \to A)\ a_0\ a_1
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$$
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%
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I will generally prefer to use the notation
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$a \equiv b$ when talking about non-dependent paths and use the notation
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$\Path\ (\lambda i \to P\ i)\ a\ b$ when the path space is of particular
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interest.
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With this definition we can also recover reflexivity. That is, for any $A \tp
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\MCU$ and $a \tp A$:
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%
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\begin{equation}
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\begin{aligned}
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\refl & \tp \Path (\lambda i \to A)\ a\ a \\
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\refl & \defeq \lambda i \to a
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\end{aligned}
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\end{equation}
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%
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Here the path space is $P \defeq \lambda i \to A$ and it satsifies $P\ i = A$
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definitionally. So to inhabit it, is to give a path $I \to A$ which is
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judgmentally $a$ at either endpoint. This is satisfied by the constant path;
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i.e.\ the path that stays at $a$ at any index $i$.
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It is also surpisingly easy to show functional extensionality with which we can
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construct a path between $f$ and $g$ -- the functions defined in the
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introduction (section \S\ref{sec:functional-extensionality}). Functional
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extensionality is the proposition, given a type $A \tp \MCU$, a family of types
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$B \tp A \to \MCU$ and functions $f, g \tp \prod_{a \tp A} B\ a$:
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%
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\begin{equation}
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\label{eq:funExt}
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\funExt \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g
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\end{equation}
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%
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%% p = λ i a → p a i
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So given $p \tp \prod_{a \tp A} f\ a \equiv g\ a$ we must give a path $f \equiv
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g$. That is a function $I \to \prod_{a \tp A} B\ a$. So let $i \tp I$ be given.
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We must now give an expression $\phi \tp \prod_{a \tp A} B\ a$ satisfying
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$\phi\ 0 \equiv f\ a$ and $\phi\ 1 \equiv g\ a$. This neccesitates that the
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expression must be a lambda-abstraction, so let $a \tp A$ be given. Now we can
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apply $a$ to $p$ and get the path $p\ a \tp f\ a \equiv g\ a$. And this exactly
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satisfied the conditions for $\phi$. In conclustion \ref{eq:funExt} is inhabited
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by the term:
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%
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\begin{equation}
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\label{eq:funExt}
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\funExt\ p \defeq λ i\ a → p\ a\ i
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\end{equation}
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%
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With this we can now prove the desired equality $f \equiv g$ from section
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\S\ref{sec:functional-extensionality}:
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%
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\begin{align*}
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p & \tp f \equiv g \\
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p & \defeq \funExt\ \lambda n \to \refl
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\end{align*}
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%
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Paths have some other important properties, but they are not the focus of
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this thesis. \TODO{Refer the reader somewhere for more info.}
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%
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\section{Homotopy levels}
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In ITT all equality proofs are identical (in a closed context). This means that,
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in some sense, any two inhabitants of $a \equiv b$ are ``equally good'' -- they
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do not have any interesting structure. This is referred to as Uniqueness of
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Identity Proofs (UIP). Unfortunately it is not possible to have a type theory
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with both univalence and UIP. In stead we have a hierarchy of types with an
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increasing amount of homotopic structure. At the bottom of this hierarchy we
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have the set of contractible types:
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%
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\begin{equation}
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\begin{aligned}
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%% \begin{split}
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& \isContr && \tp \MCU \to \MCU \\
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& \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c
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%% \end{split}
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\end{aligned}
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\end{equation}
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%
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The first component of $\isContr\ A$ is called ``the center of contraction''.
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Under the propositions-as-types interpretation of type theory $\isContr\ A$ can
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be thought of as ``the true proposition $A$''. And indeed $\top$ is
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contractible:
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\begin{equation*}
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\var{tt} , \lambda x \to \refl \tp \isContr\ \top
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\end{equation*}
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%
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It is a theorem that if a type is contractible, then it is isomorphic to the
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unit-type.
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The next step in the hierarchy is the set of mere propositions:
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%
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\begin{equation}
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\begin{aligned}
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& \isProp && \tp \MCU \to \MCU \\
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& \isProp\ A && \defeq \prod_{a_0, a_1 \tp A} a_0 \equiv a_1
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\end{aligned}
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\end{equation}
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%
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One can think of $\isProp\ A$ as the set of true and false propositions. And
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indeed both $\top$ and $\bot$ are propositions:
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%
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\begin{align*}
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λ \var{tt}\ \var{tt} → refl & \tp \isProp\ ⊤ \\
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λ\varnothing\ \varnothing & \tp \isProp\ ⊥
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\end{align*}
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%
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The term $\varnothing$ is used here to denote an impossible pattern. It is a
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theorem that if a mere proposition $A$ is inhabited, then so is it contractible.
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If it is not inhabited it is equivalent to the empty-type (or false
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proposition).\TODO{Cite!!}
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I will refer to a type $A \tp \MCU$ as a \emph{mere} proposition if I want to
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stress that we have $\isProp\ A$.
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Then comes the set of homotopical sets:
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%
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\begin{equation}
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\begin{aligned}
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& \isSet && \tp \MCU \to \MCU \\
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& \isSet\ A && \defeq \prod_{a_0, a_1 \tp A} \isProp\ (a_0 \equiv a_1)
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\end{aligned}
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\end{equation}
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%
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I will not give an example of a set at this point. It turns out that proving
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e.g.\ $\isProp\ \bN$ is not so straight-forward (see \cite[\S3.1.4]{hott-2013}).
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There will be examples of sets later in this report. At this point it should be
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noted that the term ``set'' is somewhat conflated; there is the notion of sets
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from set-theory, in Agda types are denoted \texttt{Set}. I will use it
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consistently to refer to a type $A$ as a set exactly if $\isSet\ A$ is a
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proposition.
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As the reader may have guessed the next step in the hierarchy is the type:
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%
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\begin{equation}
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\begin{aligned}
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& \isGroupoid && \tp \MCU \to \MCU \\
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& \isGroupoid\ A && \defeq \prod_{a_0, a_1 \tp A} \isSet\ (a_0 \equiv a_1)
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\end{aligned}
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\end{equation}
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%
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And so it continues. In fact we can generalize this family of types by indexing
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them with a natural number. For historical reasons, though, the bottom of the
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hierarchy, the contractible types, is said to be a \nomen{-2-type}{homotopy
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levels}, propositions are \nomen{-1-types}{homotopy levels}, (homotopical)
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sets are \nomen{0-types}{homotopy levels} and so on\ldots
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Just as with paths, homotopical sets are not at the center of focus for this
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thesis. But I mention here some properties that will be relevant for this
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exposition:
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Proposition: Homotopy levels are cumulative. That is, if $A \tp \MCU$ has
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homotopy level $n$ then so does it have $n + 1$.
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Let $\left\Vert A \right\Vert = n$ denote that the level of $A$ is $n$.
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Proposition: For any homotopic level $n$ this is a mere proposition.
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%
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\section{A few lemmas}
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Rather than getting into the nitty-gritty details of Agda I venture to take a
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more ``combinator-based'' approach. That is, I will use theorems about paths
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already that have already been formalized. Specifically the results come from
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the Agda library \texttt{cubical} (\TODO{Cite}). I have used a handful of
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results from this library as well as contributed a few lemmas myself.\footnote{The module \texttt{Cat.Prelude} lists the upstream dependencies. As well my contribution to \texttt{cubical} can be found in the git logs \TODO{Cite}.}
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These theorems are all purely related to homotopy theory and cubical Agda and as
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such not specific to the formalization of Category Theory. I will present a few
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of these theorems here, as they will be used later in chapter
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\ref{ch:implementation} throughout.
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\subsection{Path induction}
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\label{sec:pathJ}
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The induction principle for paths intuitively gives us a way to reason about a
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type family indexed by a path by only considering if said path is $\refl$ (the
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\nomen{base case}{path induction}). For \emph{based path induction}, that equality is \emph{based}
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at some element $a \tp A$.
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Let a type $A \tp \MCU$ and an element of the type $a \tp A$ be given. $a$ is said to be the base of the induction. Given a family of types:
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%
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$$
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D \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} \MCU
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$$
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%
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And an inhabitant of $D$ at $\refl$:
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%
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$$
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d \tp D\ a\ \refl
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$$
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%
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We have the function:
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%
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\begin{equation}
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\pathJ\ D\ d \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} D\ a\ p
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\end{equation}
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%
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A simple application of $\pathJ$ is for proving that $\var{sym}$ is an
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involution. Namely for any set $A \tp \MCU$, points $a, b \tp A$ and a path
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between them $p \tp a \equiv b$:
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%
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\begin{equation}
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\label{eq:sym-invol}
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\var{sym}\ (\var{sym}\ p) ≡ p
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\end{equation}
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%
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The proof will be by induction on $p$ and will be based at $a$. That is, $D$
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will be the family:
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%
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\begin{align*}
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D & \tp \prod_{b' \tp A} \prod_{p \tp a ≡ b'} \MCU \\
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D\ b'\ p' & \defeq \var{sym}\ (\var{sym}\ p') ≡ p'
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\end{align*}
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%
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The base case will then be:
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%
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\begin{align*}
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d & \tp \var{sym}\ (\var{sym}\ \refl) ≡ \refl \\
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d & \defeq \refl
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\end{align*}
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%
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The reason $\refl$ proves this is that $\var{sym}\ \refl = \refl$ holds
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definitionally. In summary \ref{eq:sym-invol} is inhabited by the term:
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%
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\begin{align*}
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\pathJ\ D\ d\ b\ p
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\tp
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\var{sym}\ (\var{sym}\ p) ≡ p
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\end{align*}
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%
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Another application of $\pathJ$ is for proving associativity of $\trans$. That
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is, given a type $A \tp \MCU$, elements of $A$, $a, b, c, d \tp A$ and paths
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between them, $p \tp a \equiv b$, $q \tp b \equiv c$ and $r \tp c \equiv d$ we
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have the following:
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%
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\begin{equation}
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\label{eq:cum-trans}
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\trans\ p\ (\trans\ q\ r) ≡ \trans\ (\trans\ p\ q)\ r
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\end{equation}
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%
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In this case the induction will be based at $c$ (the left-endpoint of $r$) and
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over the family:
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%
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\begin{align*}
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T & \tp \prod_{d' \tp A} \prod_{r' \tp c ≡ d'} \MCU \\
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T\ d'\ r' & \defeq \trans\ p\ (\trans\ q\ r') ≡ \trans\ (\trans\ p\ q)\ r'
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\end{align*}
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%
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So the base case is proven with $t$ which is defined as:
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%
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\begin{align*}
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\trans\ p\ (\trans\ q\ \refl) & ≡
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\trans\ p\ q \\
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& ≡
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\trans\ (\trans\ p\ q)\ \refl
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\end{align*}
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%
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Here we have used the proposition $\trans\ p\ \refl \equiv p$ without proof. In
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conclusion \ref{eq:cum-trans} is inhabited by the term:
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%
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\begin{align*}
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\pathJ\ T\ t\ d\ r
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\end{align*}
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%
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We shall see another application on path induction in \ref{eq:pathJ-example}.
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\subsection{Paths over propositions}
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\label{sec:lemPropF}
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Another very useful combinator is $\lemPropF$:
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To `promote' this to a dependent path we can use another useful combinator;
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$\lemPropF$. Given a type $A \tp \MCU$ and a type family on $A$; $P \tp A \to
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\MCU$. Let $\var{propP} \tp \prod_{x \tp A} \isProp\ (P\ x)$ be the proof that
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$P$ is a mere proposition for all elements of $A$. Furthermore say we have a
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path between some two elements in $A$; $p \tp a_0 \equiv a_1$ then we can built
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a heterogeneous path between any two elements of $p_0 \tp P\ a_0$ and $p_1 \tp
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P\ a_1$:
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%
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$$
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\lemPropF\ \var{propP}\ p \tp \Path\ (\lambda\; i \mto P\ (p\ i))\ p_0\ p_1
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$$
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%
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This is quite a mouthful. So let me try to show how this is a very general and
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useful result.
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Often when proving equalities between elements of some dependent types
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$\lemPropF$ can be used to boil this complexity down to showing that the
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dependent parts of the type are mere propositions. For instance, saw we have a type:
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%
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$$
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T \defeq \sum_{a \tp A} P\ a
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$$
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%
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For some proposition $P \tp A \to \MCU$. If we want to prove $t_0 \equiv t_1$
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for two elements $t_0, t_1 \tp T$ then this will be a pair of paths:
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%
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%
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\begin{align*}
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p \tp & \fst\ t_0 \equiv \fst\ t_1 \\
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& \Path\ (\lambda i \to P\ (p\ i))\ \snd\ t_0 \equiv \snd\ t_1
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\end{align*}
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%
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Here $\lemPropF$ directly allow us to prove the latter of these:
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%
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$$
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\lemPropF\ \var{propP}\ p
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\tp \Path\ (\lambda i \to P\ (p\ i))\ \snd\ t_0 \equiv \snd\ t_1
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$$
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%
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\subsection{Functions over propositions}
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\label{sec:propPi}
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$\prod$-types preserve propositionality when the co-domain is always a
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proposition.
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%
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$$
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\mathit{propPi} \tp \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\prod_{a \tp A} P\ a\right)
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$$
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\subsection{Pairs over propositions}
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\label{sec:propSig}
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%
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$\sum$-types preserve propositionality whenever its first component is a
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proposition, and its second component is a proposition for all points of in the
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left type.
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%
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$$
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\mathit{propSig} \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right)
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$$
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