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\tableofcontents
\section{Idea}
Dependent type theory is the foundations of mathematics.
\section{Presentation}
The model of dependent type theory we shall be presenting here is the objective type theory version of dependent type theory. There are multiple reasons for this:
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Since objective type theory lacks definitional equality,
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The ruleset is simpler in the objective type theory model of homotopy type theory than other models such as Martin-Löf type theory, cubical type theory, or higher observational type theory
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The results in objective type theory are more general than in models which use definitional equality
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It is similar to other non-type theory foundations such as the various flavors of set theory, since it also only has one notion of equality, which is propositional equality in both objective type theory and set theory, and uses propositional equality to define terms and types.
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From a more practical standpoint, objective type theory not only has decidable type checking, it has polynomial (quadratic) time type checking, which makes it efficient from a computational standpoint.
\subsection{Judgments and contexts}
Objective type theory consists of three judgments: type judgments , where we judge to be a type, typing judgments, where we judge to be an element of , , and context judgments, where we judge to be a context, . Contexts are lists of typing judgments , , , et cetera, and are formalized by the rules for the empty context and extending the context by a typing judgment
\subsection{Structural rules}
There are three structural rules in objective type theory, the variable rule?, the weakening rule?, and the substitution rule?.
The variable rule states that we may derive a typing judgment if the typing judgment is in the context already:
Let be any arbitrary judgment. Then we have the following rules:
The weakening rule:
The substitution rule:
The weakening and substitution rules are admissible rules: they do not need to be explicitly included in the type theory as they could be proven by induction on the structure of all possible derivations.
There are other rules which could be derived from the three rules above. These include the exchange rule and the variable conversion rule.
The exchange rule:
\subsection{Dependent types and sections}
A dependent type is a type in the context of the variable judgment , , they are usually written as to indicate its dependence upon .
A section or dependent term is a term in the context of the variable judgment , . Sections are likewise usually written as to indicate its dependence upon .
\subsection{Equality}
Equality of elements of a type in objective type theory is represented by a type known as the equality type or the type of equalities. The elements of the equality type are called equalities.
Equality comes with a formation rule, an introduction rule, an elimination rule, and a computation rule:
Formation rule for equality types:
Introduction rule for equality types:
Elimination rule for equality types:
Computation rules for equality types:
Optional uniqueness rules for equality types:
The uniqueness rule for equality types is usually not included in objective type theory. However, if it were included in objective type theory it turns the type theory into a set-level type theory?.
Structural rules for definitions
Now that we have finally defined equality types, we can define the structural rules for definitions. The structural rules for term definitions say that given a term and a term definition , one could derive that is a term of , and that there is an equality between and :
The structural rules for type definitions are the natural deduction rules for copying? types, with formation and introduction rules:
elimination rules:
computation rules:
and uniqueness rules:
Function types
Formation rules for function types:
Introduction rules for function types:
Elimination rules for function types:
Computation rules for function types:
Uniqueness rules for function types:
Pi types
Formation rules for Pi types:
Introduction rules for Pi types:
Elimination rules for Pi types:
Computation rules for Pi types:
Uniqueness rules for Pi types:
Product types
We use the negative presentation for product types.
Formation rules for product types:
Introduction rules for product types:
Elimination rules for product types:
Computation rules for product types:
Uniqueness rules for product types:
Sigma types
We use the negative presentation for sigma types.
Formation rules for Sigma types:
Introduction rules for Sigma types:
Elimination rules for Sigma types:
Computation rules for Sigma types:
Uniqueness rules for Sigma types:
Sum types
Formation rules for sum types:
Introduction rules for sum types:
Elimination rules for sum types:
Computation rules for sum types:
Uniqueness rules for sum types:
Empty type
Formation rules for the empty type:
Elimination rules for the empty type:
Uniqueness rules for the empty type:
Unit type
Formation rules for the unit type:
Introduction rules for the unit type:
Elimination rules for the unit type:
Computation rules for the unit type:
Uniqueness rules for the unit type:
Booleans
Formation rules for the booleans:
Introduction rules for the booleans:
Elimination rules for the booleans:
Computation rules for the booleans:
Uniqueness rules for the booleans:
\subsection{Natural numbers}
The natural numbers are the first type introduced in mathematics, so it would be appropriate to introduce them here as well.
Formation rules for the natural numbers:
Introduction rules for the natural numbers:
Elimination rules for the natural numbers:
Computation rules for the natural numbers:
Uniqueness rules for the natural numbers:
Equivalence types
Given a type , we define the type representing whether is a contractible type as
Given types and , function , and element , we define the fiber of at as
Given types and and function , we define the type representing whether is a equivalence of types? as
Given types and , we define the type of equivalences as
The equivalence types between two types and behaves as the equality between and , in the same way that the identity type between two terms and behaves as the equality between and . This is similar to structural set theory? whose type of sets have no primitive equality relation, where bijection? behaves as the equality between sets and .
Transport
Transport is the statement that given a type family indexed by and elements and , there is a function from the identity type of and to the type of equivalences between the types and . This is inductively defined on reflexivity on , which transport takes to the identity function on , .
Transport is given by the following rules:
Transport is very important in defining univalent Tarski universes? as well as dependent identity types?, which in turn are important in defining higher inductive types, in objective type theory.
Dependent identity types
We define the dependent identity type? as follows:
Dependent actions on identifications
Additionally, for a term in the context of , there is a dependent identification? called the dependent action on identifications? for all , , and , inductively defined by reflexivity for all .
The rules for are as follows
\section{See also}
\section{References}
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Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study, 2013. (web, pdf)
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Egbert Rijke, Introduction to Homotopy Type Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press (pdf) (478 pages)
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Benno van den Berg, Martijn den Besten, Quadratic type checking for objective type theory (arXiv:2102.00905)
\section{External links}