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\begin{document}

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\section*{substitution}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{syntactic_substitution}{Syntactic substitution}\dotfill \pageref*{syntactic_substitution} \linebreak
\noindent\hyperlink{simultaneous_substitution}{Simultaneous substitution}\dotfill \pageref*{simultaneous_substitution} \linebreak
\noindent\hyperlink{avoiding_variable_capture}{Avoiding variable capture}\dotfill \pageref*{avoiding_variable_capture} \linebreak
\noindent\hyperlink{as_an_admissible_rule}{As an admissible rule}\dotfill \pageref*{as_an_admissible_rule} \linebreak
\noindent\hyperlink{explicit_substitution}{Explicit substitution}\dotfill \pageref*{explicit_substitution} \linebreak
\noindent\hyperlink{CategoricalSemantics}{Categorical semantics}\dotfill \pageref*{CategoricalSemantics} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Syntactic \emph{substitution} for [[variables]] is one of the basic operations in formal mathematics, such as symbolic [[logic]] and [[type theory|type theories]].

\hypertarget{syntactic_substitution}{}\subsection*{{Syntactic substitution}}\label{syntactic_substitution}

Substitution is usually defined as an \emph{operation} on expressions containing variables. (These expressions may be [[terms]], [[formulas]], [[propositions]], [[dependent types]], etc.) Suppose that $P$ an expression in the [[context]] of a [[variable]] $x$, and that $t$ is an expression which has the same type as $x$. Then we denote by

\begin{displaymath}
P[t/x]
\end{displaymath}
the result of \textbf{substituting} $t$ for all occurrences of $x$ in $P$.

For example, if $P$ is $x^2 + 2x y + 3$ and $t$ is $(y+z)$, then $P[t/x]$ is $(y+z)^2 + 2(y+z)y + 3$.

Note that in this approach, substitution is an \emph{operation on syntax}, not an element of syntax itself. In particular, the bracket notation $[t/x]$ is part of ``meta-syntax'', not the syntax in question. That is, the literal string of symbols ``$P[t/x]$'' is not itself an expression in the language under consideration, but \emph{denotes} such an expression, in the same way that ``$2+2$'' is not literally an integer but \emph{denotes} the integer $4$.

\hypertarget{simultaneous_substitution}{}\subsubsection*{{Simultaneous substitution}}\label{simultaneous_substitution}

Substitution for multiple variables does not, in general, commute. That is,

\begin{displaymath}
P[t/x][s/y] \qquad\text{and}\qquad P[s/y][t/x]
\end{displaymath}
are not in general the same: the former substitutes $s$ for occurrences of $y$ in $t$, but not $t$ for occurrences of $x$ in $s$, while the latter has the opposite behavior. We also write

\begin{displaymath}
P[t,s/x,y]
\end{displaymath}
to denote the \emph{simultaneous substitution} of $t$ for $x$ and $s$ for $y$, in which \emph{neither} occurrences of $x$ in $s$ nor occurrences of $y$ in $t$ are substituted for; this is generally not the same as either iterated substitution.

\hypertarget{avoiding_variable_capture}{}\subsubsection*{{Avoiding variable capture}}\label{avoiding_variable_capture}

If the language in question contains variable [[bound variable|binders]], then there is a subtlety to substitution: if $t$ contains free variables that are bound in $P$, then we cannot simply textually substitute $t$ for $x$ and obtain an expression with the desired meaning.

For instance, if $P$ is $\exists y (x + y = 1)$, and $t$ is the free variable $y$, then a literal interpretation of $P[t/x]$ would produce $\exists y (y + y = 1)$. But $P$ is true (universally in its free variables) if the variables have type $\mathbb{Z}$, while $\exists y (y + y = 1)$ is not. The free variable $y$ in $t$ has been ``captured'' by the binder $\exists y$ in $P$.

We say that $t$ is \textbf{substitutable} for $x$ in $P$ if performing a literal textual substitution as above would not result in undesired variable capture. If $t$ is not substitutable for $x$ in $P$, then we can always replace $P$ by an [[alpha-equivalent]] expression in which $t$ \emph{is} substitutable for $x$. Since we often consider formulas only up to $\alpha$-equivalence anyway, one usually defines the notation ``$P[t/x]$'' to include an $\alpha$-conversion of $P$, if necessary, to make $t$ substitutable for $x$.

In computer implementations of type theories, however, the issue of variable binding and capture is one of the trickiest things to get right. Performing $\alpha$-conversions is difficult and tedious, and other solutions exist, such as using [[de Brujin indices]] to represent bound variables.

\hypertarget{as_an_admissible_rule}{}\subsubsection*{{As an admissible rule}}\label{as_an_admissible_rule}

A general property of type theories (and other formal mathematics) is that \emph{substitution is an admissible rule}. Roughly, this means that if $P$ is an expression of some type, then so is the result $P[t/x]$ of substitution (as long as $t$ and $x$ have the same type). This is generally not a rule ``put into'' the theory, but rather a property one \emph{proves} about the theory; type theorists say that substitution is an [[admissible rule]] rather than a [[derivable rule]].

For instance, in the language of [[dependent type theory]] we can show that the following substitution rule is admissible:

\begin{displaymath}
\frac{\Gamma \vdash (t:A) \qquad (x:A) \vdash P \;type}{\Gamma \vdash P[t/x] \;type}
\end{displaymath}
Here ``admissibility'' means that if there exist derivations of $\Gamma \vdash (t:A)$ and $(x:A) \vdash P \;type$, then there also exists a derivation of $\Gamma \vdash P[t/x] \;type$. By contrast, saying that this is a derivable rule would mean that it can occur itself as \emph{part} of a derivation, rather than being a meta-statement \emph{about} derivations.

The substition rule is closely related to the [[cut rule]], and admissibility of such rules is generally proven by [[cut elimination]].

\hypertarget{explicit_substitution}{}\subsection*{{Explicit substitution}}\label{explicit_substitution}

An alternative approach to substitution is to make substitution part of the object language rather than the metalanguage. That is, the notation

\begin{displaymath}
P[t/x]
\end{displaymath}
is now actually itself a string of the language under consideration. One then needs reduction or equality rules describing the relationship of this string $P[t/x]$ to the result of actually substituting $t$ for $x$ as in the usual approach. See [[explicit substitution]] for more details.

\hypertarget{CategoricalSemantics}{}\subsection*{{Categorical semantics}}\label{CategoricalSemantics}

\hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition}

In the [[categorical semantics]] of [[type theory]]:

\begin{itemize}%
\item Recalling that [[terms]] are interpreted by [[morphisms]], substitution of a term into another term is interpreted by [[composition]] of the relevant morphisms.


\item Recalling that [[propositions]] are interpreted by [[subobjects]], substitution of a term $t$ into a proposition $P$ is interpreted by [[pullback]] or [[inverse image]] of the subobject interpreting $P$ along the morphism interpreting $t$.


\item Recalling that [[dependent types]] are interpreted by [[display maps]], substitution of a term $t$ into a dependent type $B$ is interpreted by [[pullback]] of the display map interpreting $B$ along the morphism interpreting $t$.


\item Or else, since [[dependent types]] are also given by [[classifying morphisms]] into a [[type of types]], substitution corresponds to [[composition]] of these classifying morphisms with the given morphism.



\end{itemize}
In the third case, there is a [[coherence]] issue: syntactic substitution in the usual approach is strictly [[associativity|associative]], whereas pullback in a category is not. One way to deal with this is by using explicit substitution as described above. Another way is to strictify the category before modeling type theory; see \emph{[[categorical model of dependent types]]}. For literature see (\hyperlink{CurienGarnerHofmann}{Curien-Garner-Hofmann}, \hyperlink{LumsdaineWarren13}{Lumsdaine-Warren 13}).

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

Let $\mathcal{C}$ be a suitable ambient [[category]] in which we are [[categorical semantics|interpreting]] [[logic]]/[[type theory]].

Suppose $X$ and $Y$ are [[types]], hence interpreted as [[objects]] of $\mathcal{C}$. Then a [[term]] of [[function type]] $f : X \to Y$ is interpreted by a [[morphism]], going by the same symbols.

Now a [[proposition]] about [[terms]] of [[type]] $Y$

\begin{displaymath}
y : Y \vdash P(y)
\end{displaymath}
is interpreted as a [[object]] of the [[slice category]] $\mathcal{C}_{/Y}$, specifically as a [[truncated object of an (infinity,1)-category|(-1)-truncated]] object if it is to be a [[proposition]], hence by a [[monomorphism]]

\begin{displaymath}
\itexarray{
    P 
    \\
    \downarrow
    \\
    Y
  }
  \,.
\end{displaymath}
For instance if $\mathcal{C} =$ [[Set]] then this is the inclusion of the [[subset]] of elements of $Y$ on which $P$ is true. And generally we may write

\begin{displaymath}
P = \{y : Y | isInhab(P(y)) \}
  \,.
\end{displaymath}
Now finally the \textbf{substitution} of $f(x)$ for $y$ in $P$, hence the proposition

\begin{displaymath}
\itexarray{
    P(f(-))
    \\
    \downarrow
    \\
    X
  }
\end{displaymath}
is interpreted as the [[pullback]]

\begin{displaymath}
\itexarray{
    P(f(-)) \coloneqq & f^* P &\to& P
    \\
    & \downarrow && \downarrow
    \\
    & X &\stackrel{f}{\to}& Y
  }
  \,.
\end{displaymath}
Notice that [[monomorphisms]] are preserved by pullback, so that this is indeed again the correct interpretation of a proposition.

Specifically, if $X$ is the [[unit]] type it is interpreted as a [[terminal object]] of $\mathcal{C}$, and then the [[function]] $f$ is identified simply with a [[term]] $y_0 \coloneqq f(*)$. In this case the substitution is \emph{evaluation} of the proposition at $y_0$, the resulting monomorphism

\begin{displaymath}
\itexarray{
    P(y_0) &\to& P
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{y_0}{\to}& Y
  }
\end{displaymath}
over the terminal object is a [[truth value]]: the truth value of $P$ at $y_0$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include substitution natural deduction - table]]

\begin{itemize}%
\item [[context extension]]


\item [[dependent product]], [[universal quantifier]]


\item [[dependent sum]], [[existential quantifier]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The observation that substitution forms an [[adjoint pair]]/[[adjoint triple]] with [[quantifiers]] is due to

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Adjointness in Foundations}, (\href{http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html}{TAC}), Dialectica 23 (1969), 281-296

\end{itemize}
and further developed in

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Equality in hyperdoctrines and comprehension schema as an adjoint functor}, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

\end{itemize}
Exposition of the interpretation of substitution as pullback is in

\begin{itemize}%
\item [[Andrej Bauer]], \emph{\href{http://math.andrej.com/2012/09/28/substitution-is-pullback/}{Substitution is pullback}}, 2012

\end{itemize}
The [[coherence]] issue involved in making this precise is discussed in

\begin{itemize}%
\item [[Pierre-Louis Curien]], [[Richard Garner]], [[Martin Hofmann]], \emph{Revisiting the categorical interpretation of dependent type theory} ([[CurienGarnerHofmann.pdf:file]])


\item [[Peter LeFanu Lumsdaine]], [[Michael Warren]], \emph{An overlooked coherence construction for dependent type theory}, CT2013 ([[LumsdaineWarren2013.pdf:file]])



\end{itemize}
[[!redirects substitution]] [[!redirects substitutions]] [[!redirects variable substitution]] [[!redirects variable substitutions]]

[[!redirects variable capture]] [[!redirects variable captures]]



\end{document}