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\section*{Martin-Löf dependent type theory}

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

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

[[!include type theory - contents]]

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\hypertarget{martinlf_dependent_type_theory}{}\section*{{Martin-L\"o{}f dependent type theory}}\label{martinlf_dependent_type_theory}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{syntax}{Syntax}\dotfill \pageref*{syntax} \linebreak
\noindent\hyperlink{binary_product_types}{Binary product types}\dotfill \pageref*{binary_product_types} \linebreak
\noindent\hyperlink{unit_type}{Unit type}\dotfill \pageref*{unit_type} \linebreak
\noindent\hyperlink{dependent_product_types}{Dependent product types}\dotfill \pageref*{dependent_product_types} \linebreak
\noindent\hyperlink{function_types}{Function types}\dotfill \pageref*{function_types} \linebreak
\noindent\hyperlink{binary_sum_types}{Binary sum types}\dotfill \pageref*{binary_sum_types} \linebreak
\noindent\hyperlink{empty_type}{Empty type}\dotfill \pageref*{empty_type} \linebreak
\noindent\hyperlink{dependent_sum_types}{Dependent sum types}\dotfill \pageref*{dependent_sum_types} \linebreak
\noindent\hyperlink{finite_types}{Finite types}\dotfill \pageref*{finite_types} \linebreak
\noindent\hyperlink{propositions_as_types}{Propositions as types}\dotfill \pageref*{propositions_as_types} \linebreak
\noindent\hyperlink{equality_types}{Equality types}\dotfill \pageref*{equality_types} \linebreak
\noindent\hyperlink{finite_product_types}{Finite product types}\dotfill \pageref*{finite_product_types} \linebreak
\noindent\hyperlink{unit_type_2}{Unit type}\dotfill \pageref*{unit_type_2} \linebreak
\noindent\hyperlink{dependent_product_types_2}{Dependent product types}\dotfill \pageref*{dependent_product_types_2} \linebreak
\noindent\hyperlink{function_types_2}{Function types}\dotfill \pageref*{function_types_2} \linebreak
\noindent\hyperlink{finite_sum_types}{Finite sum types}\dotfill \pageref*{finite_sum_types} \linebreak
\noindent\hyperlink{dependent_sum_types_2}{Dependent sum types}\dotfill \pageref*{dependent_sum_types_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{models_and_categorical_semantics}{Models and categorical semantics}\dotfill \pageref*{models_and_categorical_semantics} \linebreak
\noindent\hyperlink{AxiomOfChoice}{Axiom of choice}\dotfill \pageref*{AxiomOfChoice} \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}

[[Per Martin-Lof|Per Martin-Löf]]`s \textbf{[[dependent type theory]]}, also known as \textbf{[[intuitionistic logic|intuitionistic]] type theory}, or \textbf{constructive type theory} is a specific form of [[type theory]] developed to support [[constructive mathematics]]. (Note that both ``dependent type theory'' and ``intuitionistic type theory'' may refer more generally to [[type theories]] that contain [[dependent types]] or are [[intuitionistic mathematics|intuitionistic]], respectively.)

Martin-L\"o{}f's dependent type theory is notable for several reasons:

\begin{itemize}%
\item One can construct an interpretation of first-order [[intuitionistic logic]] by interpreting [[propositions as types]] (this is true of most any dependent type theory).


\item It has a version of a variant of the [[axiom of choice]] \emph{as a theorem} (because of the properties of the above interpretation), see the discussion \hyperlink{AxiomOfChoice}{below}.


\item In its [[intensional type theory|intensional]] form, it has sufficient computational content to function as a programming language. At the same time, it then has [[identity types]] whose presence shows that one is really dealing with a form of [[homotopy type theory]].



\end{itemize}
\hypertarget{syntax}{}\subsection*{{Syntax}}\label{syntax}

If $t$ is a term, and $A$ a type, then $t : A$ is a \textbf{typing declaration} asserting that $t$ is a term of type $A$. We will define the forms of valid terms and types below; to begin with, we assume we have a stock of variables, atomic types, and parametrised types.

A \textbf{context} is a [[finite set|finite]] ordered list of typing declarations, defined [[inductive definition|inductively]] as follows:

\begin{itemize}%
\item The empty list is a valid context.
\item If $\Gamma$ is a valid context, $x$ is a variable not appearing in $\Gamma$, and $A$ is a valid type in the context $\Gamma$, then the context obtained by appending $x : A$ to $\Gamma$ is also valid.

\end{itemize}
Note that this must be considered a \emph{mutual} inductive definition along with the definition of when a type is valid in a given context, which is to be given below in terms of type constructors.

We write $\Gamma \vdash t : A$ to assert that $t : A$ is a valid typing declaration in the context of $\Gamma$, and by abuse of notation we may write $\Gamma \vdash A : \mathrm{Type}$ to assert that $A$ is a valid type in the context of $\Gamma$.

For any context $\Gamma$, if $A$ is an atomic type, $\Gamma \vdash A : \mathrm{Type}$.

If $x : A$ is a typing declaration in $\Gamma$, then we have $\Gamma \vdash x : A$.

\hypertarget{binary_product_types}{}\subsubsection*{{Binary product types}}\label{binary_product_types}

If $\Gamma \vdash A : \mathrm{Type}$ and $\Delta \vdash B : \mathrm{Type}$, then $\Gamma, \Delta \vdash A \times B : \mathrm{Type}$.

If $\Gamma \vdash a : A$ and $\Delta \vdash b : B$ then $\Gamma, \Delta \vdash \langle a, b \rangle : A \times B$.

If $\Gamma \vdash c : A \times B$, then $\Gamma \vdash \pi_0(c) : A$ and $\Gamma \vdash \pi_1(c) : B$.

\hypertarget{unit_type}{}\subsubsection*{{Unit type}}\label{unit_type}

There is an atomic type called $\top$.

For any context $\Gamma$ we have $\Gamma \vdash \mathrm{tt} : \top$.

\hypertarget{dependent_product_types}{}\subsubsection*{{Dependent product types}}\label{dependent_product_types}

If $\Gamma, x : X \vdash A(x) : \mathrm{Type}$, then $\Gamma \vdash (\Pi x : X) A(x) : \mathrm{Type}$.

If $\Gamma, x : X \vdash a(x) : A(x)$, then $\Gamma \vdash (\lambda x : X) a(x) : (\Pi x : X) A(x)$.

If $\Gamma \vdash f : (\Pi x : X) A(x)$ and $\Delta \vdash t : X$, then $\Gamma, \Delta \vdash \mathrm{apply}(f, t) : A(t)$.

\hypertarget{function_types}{}\subsubsection*{{Function types}}\label{function_types}

Function types may be regarded as a special case of dependent product types, where $A(x)$ does not depend on $x$ at all. When we write out the rules for dependent products in this case, they become the following.

Firstly, if $\Gamma \vdash X : \mathrm{Type}$ and $\Gamma \vdash A : \mathrm{Type}$, then $\Gamma \vdash X \to A : \mathrm{Type}$.

If $\Gamma, x : X \vdash a(x) : A$, then $\Gamma \vdash (\lambda x : X) a(x) : X \to A$.

If $\Gamma \vdash f : X \to A$ and $\Delta \vdash t : X$, then $\Gamma, \Delta \vdash \mathrm{apply}(f, t) : A$.

\hypertarget{binary_sum_types}{}\subsubsection*{{Binary sum types}}\label{binary_sum_types}

If $\Gamma \vdash A : \mathrm{Type}$ and $\Gamma \vdash B : \mathrm{Type}$, then $\Gamma \vdash A + B : \mathrm{Type}$.

If $\Gamma \vdash a : A$ and $\Gamma \vdash A + B : \mathrm{Type}$, then $\Gamma \vdash a : A + B$; and if $\Gamma \vdash b : B$ and $\Gamma \vdash A + B : \mathrm{Type}$, then $\Gamma \vdash b : A + B$.

If $\Gamma \vdash s : A + B$, $\Delta, x : A \vdash c(x) : C(x)$ and $E, y : B \vdash c'(y) : C(y)$, then $\Gamma, \Delta, E \vdash \mathrm{cases}(s, (\lambda x : A) c(x), (\lambda y : B) c'(y)) : C(s)$.

\hypertarget{empty_type}{}\subsubsection*{{Empty type}}\label{empty_type}

There is an atomic type called $\bot$.

If $\Gamma \vdash A : \mathrm{Type}$, then $\Gamma, x : \bot \vdash \mathrm{abort} : A$.

\hypertarget{dependent_sum_types}{}\subsubsection*{{Dependent sum types}}\label{dependent_sum_types}

If $\Gamma, x : X \vdash A(x) : \mathrm{Type}$, then $\Gamma \vdash (\Sigma x : X) A(x) : \mathrm{Type}$.

If $\Gamma \vdash t : X$, $\Gamma \vdash a : A(t)$ and $\Gamma \vdash (\Sigma x : X) A(x) : \mathrm{Type}$, then $\Gamma \vdash (t, a) : (\Sigma x : X) A(x)$.

If $\Gamma \vdash s : (\Sigma x : X) A(x)$ and $\Delta, x : X, y : A(x) \vdash b(x, y) : B((x, y))$, then $\Gamma, \Delta \vdash \mathrm{cases}(s, (\lambda x : X)(\lambda y : A(x)) b(x, y)) : B(s)$.

Note that just as function types can be defined to be dependent products where $A(x)$ does not depend on $x$, binary product types (above) can be defined to be dependent sums where $A(x)$ does not depend on $x$.

\hypertarget{finite_types}{}\subsubsection*{{Finite types}}\label{finite_types}

Martin-L\"o{}f also includes in his type theory a type $\mathbb{N}_n$ with exactly $n$ elements, for every \emph{external} [[natural number]] $n$. The types $\bot$ and $\top$ can then be defined as $\mathbb{N}_0$ and $\mathbb{N}_1$, respectively. On the other hand, with $\bot$ and $\top$ given as above, one may define $\mathbb{N}_2 = \top + \top$, $\mathbb{N}_3 = \top+\top+\top$, and so on (by [[recursion]] on the external natural number $n$).

Note that we have not included any [[axiom of infinity]], however.

\hypertarget{propositions_as_types}{}\subsubsection*{{Propositions as types}}\label{propositions_as_types}

Under the [[Curry-Howard isomorphism]], we may identify propositions with certain atomic types and predicates with certain parametrised types.

An inhabitant of such a [[propositions as types|proposition-as-a-type]] is interpreted as a `proof' of that proposition.

We write $\Gamma \vdash A : \mathrm{true}$ for the judgement that $A$ is [[inhabited set|inhabited]]: that is, if $\Gamma \vdash a : A$, then $\Gamma \vdash A : \mathrm{true}$. The type-formation rules above are then seen to be the rules of inference for (a fragment of) [[intuitionistic logic|intuitionistic]] [[first-order logic]]. Indeed, we have:

\begin{itemize}%
\item [[logical conjunction|Conjunction]] introduction:\begin{displaymath}
\frac{\Gamma \vdash A : \mathrm{true} ; \Delta \vdash B : \mathrm{true}}{\Gamma, \Delta \vdash A \times B : \mathrm{true}}
\end{displaymath}

\item Conjunction elimination:\begin{displaymath}
\begin{aligned}
\frac{\Gamma \vdash A \times B : \mathrm{true}}{\Gamma \vdash A : \mathrm{true}} \,\, &
\frac{\Gamma \vdash A \times B : \mathrm{true}}{\Gamma \vdash B : \mathrm{true}} 
\end{aligned}
\end{displaymath}

\item [[truth|Truth]] introduction\begin{displaymath}
\frac{}{\Gamma \vdash \top : \mathrm{true}}
\end{displaymath}

\item [[universal quantifier|Universal]] generalisation:\begin{displaymath}
\frac{\Gamma, x : X \vdash A(x) : \mathrm{true}}{\Gamma \vdash (\Pi x : X) A(x) : \mathrm{true}}
\end{displaymath}

\item Universal instantiation:\begin{displaymath}
\frac{\Gamma \vdash (\Pi x : X) A(x) : \mathrm{true} ; \Delta \vdash t : X}{\Gamma, \Delta \vdash A(t) : \mathrm{true}}
\end{displaymath}

\item Conditional proof ([[implication]] introduction):\begin{displaymath}
\frac{\Gamma, X : \mathrm{true} \vdash A : \mathrm{true}}{\Gamma \vdash X \to A : \mathrm{true}}
\end{displaymath}

\item Modus ponens ([[implication]] elimination):\begin{displaymath}
\frac{\Gamma \vdash X \to A : \mathrm{true} ; \Delta \vdash X : \mathrm{true}}{\Gamma, \Delta \vdash A : \mathrm{true}}
\end{displaymath}

\item [[disjunction|Disjunction]] introduction:\begin{displaymath}
\begin{aligned}
\frac{\Gamma \vdash A : \mathrm{true}}{\Gamma \vdash A + B : \mathrm{true}} \,\, &
\frac{\Gamma \vdash B : \mathrm{true}}{\Gamma \vdash A + B : \mathrm{true}}
\end{aligned}
\end{displaymath}

\item Disjunction elimination:\begin{displaymath}
\frac{\Gamma \vdash A + B : \mathrm{true} ; \Delta, A : \mathrm{true} \vdash C : \mathrm{true} ; E, B : \mathrm{true} \vdash C : \mathrm{true}}{\Gamma, \Delta, E \vdash C : \mathrm{true}}
\end{displaymath}

\item [[false|False]] elimination:\begin{displaymath}
\frac{\Gamma \vdash \bot : \mathrm{true}}{\Gamma \vdash A : \mathrm{true}}
\end{displaymath}

\item [[existential quantifier|Existential]] generalisation:\begin{displaymath}
\frac{\Gamma \vdash t : X ; \Gamma \vdash A(t) : \mathrm{true}}{\Gamma \vdash (\Sigma x : X) A(x) : \mathrm{true}}
\end{displaymath}

\item Existential instantiation:\begin{displaymath}
\frac{\Gamma \vdash (\Sigma x : X) A(x) : \mathrm{true} ; \Delta, x : X, A(x) : \mathrm{true} \vdash B : \mathrm{true}}{\Gamma, \Delta \vdash B : \mathrm{true}}
\end{displaymath}


\end{itemize}
The only traditional rule of inference we are missing is the rule of [[excluded middle]], so this logic is [[intuitionistic logic|intuitionistic]].

The above interpretation justifies the use of the symbols $\forall$, $\exists$, $\wedge$, $\vee$ instead of $\Pi$, $\Sigma$, $\times$, $+$, when we are using types to represent propositions.

\hypertarget{equality_types}{}\subsubsection*{{Equality types}}\label{equality_types}

We may introduce a family of equality types for each type and each pair of terms of that type: if $\Gamma \vdash a : A$ and $\Delta \vdash a' : A$, then $\Gamma \vdash a = a' : \mathrm{Type}$.

These are not the intensional [[identity types]] which Martin-L\"o{}f later introduced as a particular sort of [[inductive type|inductive family of types]], but instead an extensional notion of equality defined by [[induction]] \emph{over} the class of all types (regarded as inductively defined by the above type-formation clauses). Note that it is essential, for a purpose such as this, that the type-formation clauses be regarded as an inductive definition, rather than as ``operations'' defined on some pre-existing collection of types: in order to define equality in the way we are about to do, we have to be able to inspect a given type and decide uniquely which rule was used to construct it.

The equality type is reflexive, symmetric, and transitive. That means, for example, if $\Gamma \vdash a : A$, $\Gamma \vdash b : A$, $\Gamma \vdash c : A$, $\Gamma \vdash a = b : \mathrm{true}$ and $\Gamma \vdash b = c : \mathrm{true}$ then $\Gamma \vdash a = c : \mathrm{true}$.

[[Mike Shulman]]: Is that (reflexivity, symmetry, and transitivity) a rule we are giving, or an assertion we are making about derivability? I'm a little unsure because the presentation here is slightly different from that in the reference: there Martin-L\"o{}f uses both a propositional equality and a judgmental one.

\hypertarget{finite_product_types}{}\subsubsection*{{Finite product types}}\label{finite_product_types}

If $\Gamma \vdash a : A$, and $\Gamma \vdash b : B$, then $\Gamma \vdash a = \pi_0 (\langle a, b \rangle) : \mathrm{true}$ and $\Gamma \vdash \pi_1 (\langle a, b \rangle) = b : \mathrm{true}$.

If $\Gamma \vdash c : A \times B$, then $c = \langle \pi_0 (c), \pi_1 (c) \rangle : \mathrm{true}$.

\hypertarget{unit_type_2}{}\subsubsection*{{Unit type}}\label{unit_type_2}

If $\Gamma \vdash a : \top$, then $\Gamma \vdash a = \mathrm{tt} : \mathrm{true}$.

\hypertarget{dependent_product_types_2}{}\subsubsection*{{Dependent product types}}\label{dependent_product_types_2}

If $\Gamma, x : X \vdash a(x) : A(x)$ and $\Delta \vdash t : X$, then $\Gamma, \Delta \vdash \mathrm{apply}((\lambda x : X) a(x), t) = a(t) : \mathrm{true}$.

If $\Gamma \vdash f : (\Pi x : X) A(x)$, then $\Gamma \vdash f = (\lambda x : X) \mathrm{apply}(f, x) : \mathrm{true}$.

\hypertarget{function_types_2}{}\subsubsection*{{Function types}}\label{function_types_2}

If $\Gamma, x : X \vdash a(x) : A$ and $\Delta \vdash t : X$, then $\Gamma, \Delta \vdash \mathrm{apply}((\lambda x : X) a(x), t) = a(t) : \mathrm{true}$.

If $\Gamma \vdash f : X \to A$, then $\Gamma \vdash f = (\lambda x : X) \mathrm{apply}(f, x) : \mathrm{true}$.

\hypertarget{finite_sum_types}{}\subsubsection*{{Finite sum types}}\label{finite_sum_types}

If $\Gamma \vdash a : A$ and $\Gamma \vdash A + B : \mathrm{Type}$, $\Delta, x : A \vdash c(x) : C(x)$ and $E, y : B \vdash c'(y) : C(y)$, then $\Gamma, \Delta, E \vdash \mathrm{cases}(a, (\lambda x : A) c(x), (\lambda y : B) c(y)) = c(a) : \mathrm{true}$ and $\Gamma, \Delta, E \vdash \mathrm{cases}(b, (\lambda x : A) c(x), (\lambda y : B) c'(y)) = c'(b) : \mathrm{true}$

\hypertarget{dependent_sum_types_2}{}\subsubsection*{{Dependent sum types}}\label{dependent_sum_types_2}

If $\Gamma \vdash t : X$, $\Gamma \vdash a : A(t)$ and $\Delta, x : X, y : A(x) \vdash b(x, a) : B((x, a))$, then $\Gamma, \Delta \vdash \mathrm{cases}((t, a), (\lambda x : X)(\lambda y : A(x)) b(x, y)) = b(t, a) : \mathrm{true}$.

The other equality rule for dependent sum types is derivable from the above.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{models_and_categorical_semantics}{}\subsubsection*{{Models and categorical semantics}}\label{models_and_categorical_semantics}

The [[models]] of ML type theory depend crucially on whether one considers the variant of [[extensional type theory]] or that of [[intensional type theory]].

The models of the extensional version are (just) [[locally cartesian closed categories]].

The faithful models of the intensional version with [[identity types]] are however certain [[(∞,1)-categories]], notably [[(∞,1)-toposes]], presented by [[simplicial model categories]] (\hyperlink{Warren}{Warren}). For this reason one speaks of \emph{[[homotopy type theory]]}.

For a more detailed discussion of these matters see at \emph{[[relation between type theory and category theory]]}.

\hypertarget{AxiomOfChoice}{}\subsubsection*{{Axiom of choice}}\label{AxiomOfChoice}

In dependent type theory we can verify the following ``logical form of the [[axiom of choice]]'' (\hyperlink{Bell}{Bell}, \hyperlink{Tait}{Tait}), see also (\hyperlink{Rijke}{Rijke, section 2.5.1}).

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(ACL)}

\begin{displaymath}
x : A, y : B(x) \vdash C(x, y) : Type
  \;
  \vdash 
  \;
  acl
  :
  (\forall x : A) (\exists y : B(x)) C(x, y) \to (\exists f : (\Pi x : A)B(x) )(\forall x : A) C(x, \mathrm{apply}(f, x)) 
  \,.
\end{displaymath}
\end{theorem}
One should note carefully that this ``is'' only ``the axiom of choice'' under the above propositions-as-types interpretation of the quantifiers $\forall$ and $\exists$.

\begin{remark}
\label{}\hypertarget{}{}
In the [[categorical semantics]] of this expression, using the [[propositions-as-types]] logic corresponds roughly to working with the [[subobject lattices]] in the [[ex/lex completion]] of a [[locally cartesian closed category]]; the ACL then follows since every object of the original category becomes [[projective object|projective]] in its ex/lex completion.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
If we use instead a different logic over the same type theory, such as [[bracket types]] to model the actual subobject lattices in an arbitrary [[locally cartesian closed category|lccc]] (not necessarily the ex/lex completion of anything), or the [[hProp]] logic in [[homotopy type theory]], then the ACL in that logic will not be derivable.

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

[[!include notions of type]]

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

See also the references at \emph{[[type theory]]}.

The original reference is

\begin{itemize}%
\item [[Per Martin-Löf]], \emph{Intuitionistic type theory} (1984)


\item [[Per Martin-Löf]], \emph{An intuitionistic theory of types: predicative part}, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), 73-118. (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926}{web})



\end{itemize}
A philosophical examination is

\begin{itemize}%
\item [[Göran Sundholm​]], \emph{Proof Theory and Meaning} (1986), (\href{https://openaccess.leidenuniv.nl/bitstream/handle/1887/11978/9_05?sequence=1}{pdf})

\end{itemize}
An introduction and survey (with an eye towards [[homotopy type theory]]) is in chapter 1 of

\begin{itemize}%
\item [[Michael Warren]], \emph{Homotopy theoretic aspects of constructive type theory}, PhD thesis (2008) (\href{http://www.andrew.cmu.edu/user/awodey/students/warren.pdf}{pdf})

\end{itemize}
as well as

\begin{itemize}%
\item Egbert Rijke, \emph{Homotopy type theory} (2012) (\href{http://hottheory.files.wordpress.com/2012/08/hott2.pdf}{pdf})

\end{itemize}
A discussion of ML dependent type theory as the [[internal language]] of [[locally cartesian closed categories]] is in

\begin{itemize}%
\item R. A. G. Seely, \emph{Locally cartesian closed categories and type theory}, Math. Proc. Camb. Phil. Soc. (1984) 95 (\href{http://www.math.mcgill.ca/rags/LCCC/LCCC.pdf}{pdf})

\end{itemize}
Discussion of the logical [[axiom of choice]] in dependent type theory is in

\begin{itemize}%
\item John Bell, \emph{The Axiom of choice in type theory}, Stanford Encyclopedia of philosophy, 2008 (\href{http://plato.stanford.edu/entries/axiom-choice/choice-and-type-theory.html}{web})

\end{itemize}
\begin{itemize}%
\item Tait (1994)

\end{itemize}
[[!redirects Martin-Lof dependent type theory]] [[!redirects Martin-Löf dependent type theory]] [[!redirects Martin-Lof type theory]] [[!redirects Martin-Löf type theory]] [[!redirects Martin-Löf type theories]] [[!redirects constructive type theory]] [[!redirects intuitionistic type theory]]



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