Martin-Löf dependent type theory

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## Idea

[[Per Martin-Lof|Per Martin-Löf]]'s **[[dependent type theory]]**, also known as **[[intuitionistic logic|intuitionistic]] type theory** ([Martin-Löf 75](#MartinLof75)), or **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-Löf's dependent type theory is notable for several reasons:

* One can construct an interpretation of first-order [[intuitionistic logic]] by interpreting [[propositions as types]] (this is true of most any dependent type theory).

* It has a version of a variant of the [[axiom of choice]] _as a theorem_ (because of the properties of the above interpretation), see the discussion [below](#AxiomOfChoice).

* 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]].

## Formal presentations

There are many different ways to present Martin-Löf dependent type theory in a formal system. In this article, we give two formal presentations of Martin-Löf dependent type theory in the framework of [[natural deduction]], one which has [[judgmental equality]] and only one level, and another which has a very weak [[logic]] over the [[dependent type theory]] as well as [[propositional equality]].

### Presentation 1

This first presentation uses [[judgmental equality]], and could be contrasted with the second presentation, which uses [[propositional equality]] and a very weak [[logic]] over [[dependent type theory]].

#### Judgments and contexts
 {#JudgmentsAndContexts}

The [[syntax]] of Martin-Löf dependent type theory can be constructed in two stages:

* The first is the *raw* or *untyped* syntax of the theory consisting of expressions that are readable but not necessarily meaningful.

* The second stage consists of defining the *derivable judgements* of the type theory [[induction|inductively]] which then pick out the meaningful [[contexts]], [[types]] and [[terms]].

A [[context]] is a [[list]] of succesively [[dependent types]]. The [[variables]] may be defined as [[de Bruijn indices]], in which case the type of a variable $n$ is given by $n$th type in a [[context]].

One may also define contexts as coming with a variable name, in which case one needs a notion of $\alpha$-equivalence (syntactic identity modulo renaming of bound variables) and of capture-free substitution. De Bruijn indices avoid this step but can be more obfuscating.

Types and terms are built inductively from various constructors. Types, terms and contexts are defined mutually.

We have the basic [[judgement]] forms:

* $\Gamma \; \mathrm{ctx}$

  saying that: "$\Gamma$ is a well-formed context";

* $\Gamma \vdash A\; \mathrm{type}$

  saying that "$A$ is a well-typed [[dependent type]] in [[context]] $\Gamma$";

* $\Gamma \vdash a : A$

  saying that "$a$ is a well-typed [[term]] of [[type]] $A$ in [[context]] $\Gamma$";

The following table indicates the corresponding [[categorical semantics]] [[categorical semantics of dependent types|of these notions of dependent type theory]]:

We work in the [[logical framework]] of [[natural deduction]], where everything is given by [[inference rules]].

To start with, [[contexts]] are built from [[dependent types]] by the following rules (cf. [[type telescope]]):

$$
  \frac{}{() \; \mathrm{ctx}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash A \; \mathrm{type}}{(\Gamma, a:A) \; \mathrm{ctx}}
$$

#### Structural rules
 {#StructuralRules}

Among the *[[structural rules]]* of dependent type theory are (e.g. [Jacobs (1998), p. 122](#Jacobs98), [UFP13, A.2.2](#UFP13), [Rijke (2018), Sec. 1.4](#Rijke18)):

* the [[variable rule]],

* the [[weakening rule]],

* the [[substitution rule]]

shown in the following table together with their [[categorical semantics]] [[categorical semantics of dependent types|of dependent types]]:

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.

#### Definitional equality

In addition, there are judgments for definitional equality of types and of terms:

* $\Gamma \vdash A \equiv A' \; \mathrm{type}$ - $A$ and $A'$ are judgementally equal well-typed types in context $\Gamma$.
* $\Gamma \vdash a \equiv a' : A$ - $a$ and $a'$ are judgementally equal well-typed terms of type $A$ in context $\Gamma$.

Definitional equality has its own structural rules: reflexivity, symmetry, transitivity, the principle of substitution, and the variable conversion rule.

* Reflexivity of definitional equality

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A \equiv A \; \mathrm{type}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a \equiv a:A}$$

* Symmetry of definitional equality
$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma \vdash B \equiv A \; \mathrm{type}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A}{\Gamma \vdash b \equiv a:A}$$

* Transitivity of definitional equality
$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma \vdash B \equiv C \; \mathrm{type} }{\Gamma \vdash A \equiv C \; \mathrm{type}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A \quad b \equiv c:A }{\Gamma \vdash a \equiv c:A}$$

* Principle of substitution of definitionally equal terms:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B \; \mathrm{type}}{\Gamma, \Delta[b/x] \vdash B[a/x] \equiv B[b/x] \; \mathrm{type}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c:B}{\Gamma, \Delta[b/x] \vdash c[a/x] \equiv c[b/x]: B[b/x]}$$

* Variable conversion rule:
$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}$$

#### Definitions

In addition, there are judgments for the initialization operator for types and for terms:

* $\Gamma \vdash A' \coloneqq A \; \mathrm{type}$ - $A'$ is defined to be the type $A$ in context $\Gamma$.
* $\Gamma \vdash a' \coloneqq a : A$ - $a'$ is defined to be the term $a:A$ of type $A$ in context $\Gamma$.

The initialization operator has its own structural rules: type formation, term introduction, and equality reflection.

* Formation and judgmental equality reflection rules for type definition:
$$\frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \; \mathrm{type}} \qquad \frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \equiv A \; \mathrm{type}}$$

* Introduction and judgmental equality reflection rules for term definition:
$$\frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b:A} \qquad \frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b \equiv a:A}$$

#### Rules for types

Each type in Martin-Löf dependent type theory comes with a type [[formation rule]], a term [[introduction rule]], a term [[elimination rule]], a [[computation rule]], and an optional [[uniqueness rule]]. The elimination rules we give here are not [[contextual elimination rules]], and the conversion rules given here are not [[contextual conversion rules]].

#### Function types

* Formation rules for function types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \to B \; \mathrm{type}}$$

* Introduction rules for function types:

$$\frac{\Gamma, x:A \vdash b(x):B}{\Gamma \vdash (x \mapsto b(x)):A \to B}$$

* Elimination rules for function types:

$$\frac{\Gamma \vdash f:A \to B \quad \Gamma \vdash a:A}{\Gamma \vdash f(a):B}$$

* Computation rules for function types:

$$\frac{\Gamma, x:A \vdash b(x):B \quad \Gamma \vdash a:A}{\Gamma \vdash (x \mapsto b(x))(a) \equiv b:B}$$

* Uniqueness rules for function types:

$$\frac{\Gamma \vdash f:A \to B}{\Gamma \vdash f \equiv (x \mapsto f(x)):A \to B}$$

#### Dependent product types

* Formation rules for dependent product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \prod_{x:A} B(x) \; \mathrm{type}}$$

* Introduction rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b(x):B(x)}{\Gamma \vdash \lambda(x:A).b(x):\prod_{x:A} B(x)}$$

* Elimination rules for dependent product types:

$$\frac{\Gamma \vdash f:\prod_{x:A} B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash f(a):B[a/x]}$$

* Computation rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \lambda(x:A).b(x)(a) \equiv b[a/x]:B[a/x]}$$

* Uniqueness rules for dependent product types:

$$\frac{\Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv \lambda(x).f(x):\prod_{x:A} B(x)}$$

#### Product types

* Formation rules for product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \times B \; \mathrm{type}}$$

* Introduction rules for product types:

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B}{\Gamma \vdash (a, b):A \times B}$$

* Elimination rules for product types:

$$\frac{\Gamma \vdash z:A \times B}{\Gamma \vdash \pi_1(z):A} \qquad \frac{\Gamma \vdash z:A \times B}{\Gamma \vdash \pi_2(z):B}$$

* Computation rules for product types:

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B}{\Gamma \vdash \pi_1(a, b) \equiv a:A} \qquad \frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B}{\Gamma \vdash \pi_2(a, b) \equiv b:B}$$

* Uniqueness rules for product types:

$$\frac{\Gamma \vdash z:A \times B}{\Gamma \vdash z \equiv (\pi_1(z), \pi_2(z)):A \times B}$$

#### Dependent sum types

* Formation rules for dependent sum types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \sum_{x:A} B(x) \; \mathrm{type}}$$

* Introduction rules for dependent sum types:

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B[a/x]}{\Gamma \vdash (a, b):\sum_{x:A} B(x)}$$

* Elimination rules for dependent sum types:

$$\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash \pi_1(z):A} \qquad \frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash \pi_2(z):B(\pi_1(z))}$$

* Computation rules for dependent sum types:

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B[a/x]}{\Gamma \vdash \pi_1(a, b) \equiv a:A} \qquad \frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B[a/x]}{\Gamma \vdash \pi_2(a, b) \equiv b:B(\pi_1(a, b))}$$

* Uniqueness rules for dependent sum types:

$$\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash z \equiv (\pi_1(z), \pi_2(z)):\sum_{x:A} B(x)}$$

#### Sum types

* Formation rules for sum types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A + B \; \mathrm{type}}$$

* Introduction rules for sum types:

$$\frac{\Gamma \vdash a:A}{\Gamma \vdash \mathrm{inl}(a):A + B} \qquad \frac{\Gamma \vdash b:B}{\Gamma \vdash \mathrm{inr}(b):A + B}$$

* Elimination rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c(x):C(\mathrm{inl}(x)) \quad \Gamma, y:B \vdash d(y):C(\mathrm{inr}(y)) \quad \Gamma \vdash e:A + B}{\Gamma \vdash \mathrm{ind}_{A + B}^C(c(x), d(y), e):C(e)}$$

* Computation rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c(x):C(\mathrm{inl}(x)) \quad \Gamma, y:B \vdash d(y):C(\mathrm{inr}(y)) \quad \Gamma \vdash a:A}{\Gamma \vdash \mathrm{ind}_{A + B}^C(c(x), d(y), \mathrm{inl}(a)) \equiv c(a):C(\mathrm{inl}(a))}$$
$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c(x):C(\mathrm{inl}(x)) \quad \Gamma, y:B \vdash d(y):C(\mathrm{inr}(y)) \quad \Gamma \vdash b:B}{\Gamma \vdash \mathrm{ind}_{A + B}^C(c(x), d(y), \mathrm{inr}(b)) \equiv d(b):C(\mathrm{inr}(b))}$$

* Uniqueness rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c(x):C(\mathrm{inl}(x)) \quad \Gamma, y:B \vdash d(y):C(\mathrm{inr}(y)) \quad \Gamma \vdash e:A + B \quad \Gamma, x:A + B \vdash u:C \quad \Gamma, a:A \vdash u(\mathrm{inl}(a)) \equiv c(a):C(\mathrm{inl}(a)) \quad \Gamma, b:B \vdash u(\mathrm{inr}(b)) \equiv d(b):C(\mathrm{inr}(b))}{\Gamma \vdash u(e) \equiv \mathrm{ind}_{A + B}^C(c(\mathrm{inl}(e)), d(\mathrm{inl}(e)), e):C(e)}$$

#### Empty type

* Formation rules for the empty type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{0} \; \mathrm{type}}$$

* Elimination rules for the empty type:

$$\frac{\Gamma, x:\mathbb{0} \vdash C \; \mathrm{type} \quad \Gamma \vdash p:\mathbb{0}}{\Gamma \vdash \mathrm{ind}_\mathbb{0}^C(p):C(p)}$$

* Uniqueness rules for the empty type:

$$\frac{\Gamma, x:\mathbb{0} \vdash C \; \mathrm{type} \quad \Gamma \vdash p:\mathbb{0} \quad \Gamma, x:\mathbb{0} \vdash u:C}{\Gamma \vdash u[p/x] \equiv \mathrm{ind}_\mathbb{0}^{C}(p):C[p/x]}$$

#### Unit type

* Formation rules for the unit type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{1} \; \mathrm{type}}$$

* Introduction rules for the unit type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash *:\mathbb{1}}$$

* Elimination rules for the unit type:

$$\frac{\Gamma, x:\mathbb{1} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_*:C[*/x] \quad \Gamma \vdash p:\mathbb{1}}{\Gamma \vdash \mathrm{ind}_\mathbb{1}^C(p, c_*):C[p/x]}$$

* Computation rules for the unit type:

$$\frac{\Gamma, x:\mathbb{1} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_*:C[*/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{1}^C(*, c_*) \equiv c_*:C[*/x]}$$

* Uniqueness rules for the unit type:

$$\frac{\Gamma, x:\mathbb{1} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_*:C[*/x] \quad \Gamma \vdash p:\mathbb{1} \quad \Gamma, x:\mathbb{1} \vdash u:C \quad \Gamma \vdash u[*/x] \equiv c_*:C[*/x]}{\Gamma \vdash u[p/x] \equiv \mathrm{ind}_\mathbb{1}^{C}(p, c_*):C[p/x]}$$

#### Natural numbers

* Formation rules for the natural numbers:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{N} \; \mathrm{type}}$$

* Introduction rules for the natural numbers:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{N}} \qquad \frac{\Gamma \vdash n:\mathbb{N}}{\Gamma \vdash s(n):\mathbb{N}}$$

* Elimination rules for the natural numbers:
$$\frac{\Gamma, x:\mathbb{N} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_0:C[0/x] \quad \Gamma, x:\mathbb{N}, c:C \vdash c_s:C[s(x)/x] \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C(n, c_0, c_s):C[n/x]}$$

* Computation rules for the natural numbers:
$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma, x:\mathbb{N}, c:C \vdash c_s:C[s(x)/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C(0, c_0, c_s) \equiv c_0:C[0/x]}$$

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma, x:\mathbb{N}, c:C \vdash c_s:C[s(x)/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C(s(n), c_0, c_s) \equiv c_s(n, \mathrm{ind}_\mathbb{N}^C(n, c_0, c_s)):C[s(n)/x]}$$

* Uniqueness rules for the natural numbers:
$$\frac{\Gamma, x:\mathbb{N} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_0:C[0/x] \quad \Gamma, x:\mathbb{N}, c:C \vdash c_s:C[s(x)/x] \quad \Gamma \vdash n:\mathbb{N} \quad \Gamma, x:\mathbb{N} \vdash u:C \quad \Gamma \vdash i_0(u):u[0/x] =_{C[0/x]} c_0 \quad \Gamma, x:\mathbb{N} \vdash i_s(u):u[s(x)/x] =_{C[s(x)/x]} c_s[u/c]}{\Gamma \vdash u[n/x] \equiv \mathrm{ind}_\mathbb{N}^{C}(p, c_0, c_s):C[n/x]}$$

#### Identity types

There is another version of equality, called the [[identity type]] or **typal equality**, because this equality is a type. The identity type is only be used for equality between terms of a type, and unless one is using [[Russell universes]] where types are represented by terms of a [[Russell universe]], identity types cannot be used for equality of types.

* Formation rule for identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A, b:A \vdash a =_A b \; \mathrm{type}}$$

* Introduction rule for identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A \vdash \mathrm{refl}_A(a) : a =_A a}$$

* Elimination rule for identity types:
$$\frac{\Gamma, a:A, b:A, p:a =_A b \vdash C \mathrm{type} \quad \Gamma, c:A \vdash t:C[c/a, c/b, \mathrm{refl}_A(c)/p]}{\Gamma, a:A, b:A, p:a =_A b \vdash J(t, a, b, p):C}$$

* Computation rules for identity types:
$$\frac{\Gamma, a:A, b:A, p:a =_A b \vdash C \; \mathrm{type} \quad \Gamma, c:A \vdash t:C[c/a, c/b, \mathrm{refl}_A(c)/p]}{\Gamma, c:A \vdash J(t, c, c, \mathrm{refl}(c)) \equiv t:C[c/a, c/b, \mathrm{refl}_A(c)/p]}$$

* Optional uniqueness rules for identity types:

...

#### Intensional and extensional type theories

In Martin-Löf dependent type theory, one makes a distinction between intensional and extensional type theories. A Martin-Löf dependent type theory is an [[extensional type theory]] if there is an equality reflection rule where from typal equality of terms, one could derive definitional equality of terms:

$$\frac{\Gamma, a:A, b:A \vdash p:a =_A b}{\Gamma, a:A, b:A \vdash a \equiv b:A}$$

The type theory is an [[intensional type theory]] if it doesn't have the equality reflection rule.

### Presentation 2

This presentation of Martin-Löf dependent type theory uses  a very weak [[logic]] over [[dependent type theory]], in the sense that while it does have a notion of [[proposition]], [[predicate]], and [[truth]], there are no [[conjunctions]], [[disjunctions]], [[implications]], [[universal quantifiers]], or [[existential quantifiers]] in the logic. Here, [[propositional equality]] plays the role of [[definitional equality]] and [[computational equality]] that [[judgmental equality]] played in the first presentation.

#### Judgments and contexts

The syntax of Martin-Löf dependent type theory can be constructed in two stages. The first is the *raw* or *untyped* syntax of the theory consisting of expressions that are readable but not meaningful. The second stage consists of defining the *derivable judgements* of the type theory inductively which then pick out the meaningful contexts, types and terms.

A context is a list of types. Variables can be defined as De Bruijn indices in which case the type of a variable $n$ is given by $n$th type in a context.

One may also define contexts as coming with a variable name, in which case one needs a notion of $\alpha$-equivalence (syntactic identity modulo renaming of bound variables) and of capture-free substitution. De Bruijn indices avoid this step but can be more obfuscating.

Types and terms are built inductively from various constructors. Types, terms and contexts are defined mutually.

We have the basic judgement forms:

* $\Gamma \vdash A\; \mathrm{type}$ - $A$ is a well-typed type in context $\Gamma$.
* $\Gamma \vdash a : A$ - $a$ is a well-typed term of type $A$ in context $\Gamma$.
* $\Gamma \vdash \phi \; \mathrm{prop}$ - $\phi$ is a well-formed proposition in context $\Gamma$
* $\Gamma \vdash \phi \; \mathrm{true}$ - $\phi$ is a well-formed true proposition in context $\Gamma$
* $\Gamma \; \mathrm{ctx}$ - $\Gamma$ is a well-formed context.

Contexts are defined by the following rules:

$$\frac{}{() \; \mathrm{ctx}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash A \; \mathrm{type}}{(\Gamma, a:A) \; \mathrm{ctx}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash \phi \; \mathrm{prop}}{(\Gamma, \phi \; \mathrm{true}) \; \mathrm{ctx}}$$

#### Structural rules

There are three structural rules in dependent 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:

$$\frac{\Gamma, a:A, \Delta \; \mathrm{ctx}}{\Gamma, a:A, \Delta \vdash a:A}$$

Let $\mathcal{J}$ be any arbitrary judgment. Then we have the following rules:

The weakening rule:

$$\frac{\Gamma, \Delta \vdash \mathcal{J} \quad \Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A, \Delta \vdash \mathcal{J}}$$

The substitution rule:

$$\frac{\Gamma \vdash a:A \quad \Gamma, b:A, \Delta \vdash \mathcal{J}}{\Gamma, \Delta[a/b] \vdash \mathcal{J}[a/b]}$$

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.

#### Definitional equality

Definitional equality of types and terms is formed by the following rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \equiv B \; \mathrm{prop}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma \vdash a \equiv_A b \; \mathrm{prop}}$$

Definitional equality has its own structural rules:  reflexivity, symmetry, transitivity, the principle of substitution, and the variable conversion rule.

* Reflexivity of definitional equality

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A \equiv A \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a \equiv_A a \; \mathrm{true}}$$

* Symmetry of definitional equality

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash A \equiv B \; \mathrm{true}}{\Gamma \vdash B \equiv A \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash a \equiv_A b \; \mathrm{true}}{\Gamma \vdash b \equiv_A a \; \mathrm{true}}$$

* Transitivity of definitional equality

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash A \equiv B \; \mathrm{true} \quad \Gamma \vdash B \equiv C \; \mathrm{true}}{\Gamma \vdash A \equiv C \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash c:A \quad \Gamma \vdash a \equiv_A b \; \mathrm{true} \quad \Gamma \vdash b \equiv_A c \; \mathrm{true}}{\Gamma \vdash a \equiv_A c \; \mathrm{true}}$$

* Principle of substitution of definitionally equal terms:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a : A \quad \Gamma \vdash b : A \quad \Gamma \vdash a \equiv_A b \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash B \; \mathrm{type}}{\Gamma, \Delta[b/x] \vdash B[a/x] \equiv B[b/x] \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a : A \quad \Gamma \vdash b : A \quad \Gamma \vdash a \equiv_A b \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash c:B}{\Gamma, \Delta[b/x] \vdash c[a/x] \equiv_{B[b/x]} c[b/x] \; \mathrm{true}}$$

* Variable conversion rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash A \equiv B \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}$$

#### Definitions

In addition, there are judgments for the initialization operator for types and for terms:

* $\Gamma \vdash A' \coloneqq A \; \mathrm{type}$ - $A'$ is defined to be the type $A$ in context $\Gamma$.
* $\Gamma \vdash a' \coloneqq a : A$ - $a'$ is defined to be the term $a:A$ of type $A$ in context $\Gamma$.

The initialization operator has its own structural rules: type formation, term introduction, and equality reflection.

* Formation and judgmental equality reflection rules for type definition:
$$\frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \; \mathrm{type}} \qquad \frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \equiv A \; \mathrm{true}}$$

* Introduction and judgmental equality reflection rules for term definition:
$$\frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b:A} \qquad \frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b \equiv_A a \; \mathrm{true}}$$

#### Rules for types

Each type in Martin-Löf dependent type theory comes with a type [[formation rule]], a term [[introduction rule]], a term [[elimination rule]], a [[computation rule]], and an optional [[uniqueness rule]]. The elimination rules we give here are not [[contextual elimination rules]], and the conversion rules given here are not [[contextual conversion rules]].

#### Function types

* Formation rules for function types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \to B \; \mathrm{type}}$$

* Introduction rules for function types:

$$\frac{\Gamma, x:A \vdash b(x):B}{\Gamma \vdash (x \mapsto b(x)):A \to B}$$

* Elimination rules for function types:

$$\frac{\Gamma \vdash f:A \to B \quad \Gamma \vdash a:A}{\Gamma \vdash f(a):B}$$

* Computation rules for function types:

$$\frac{\Gamma, x:A \vdash b(x):B \quad \Gamma \vdash a:A}{\Gamma \vdash (x \mapsto b(x))(a) \equiv_{B} b \; \mathrm{true}}$$

* Uniqueness rules for function types:

$$\frac{\Gamma \vdash f:A \to B}{\Gamma \vdash f \equiv_{A \to B} (x \mapsto f(x)) \; \mathrm{true}}$$

#### Dependent product types

* Formation rules for dependent product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \prod_{x:A} B(x) \; \mathrm{type}}$$

* Introduction rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b(x):B(x)}{\Gamma \vdash \lambda(x:A).b(x):\prod_{x:A} B(x)}$$

* Elimination rules for dependent product types:

$$\frac{\Gamma \vdash f:\prod_{x:A} B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash f(a):B[a/x]}$$

* Computation rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \lambda(x:A).b(x)(a) \equiv_{B[a/x]} b[a/x] \; \mathrm{true}}$$

* Uniqueness rules for dependent product types:

$$\frac{\Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv_{\prod_{x:A} B(x)} \lambda(x).f(x) \; \mathrm{true}}$$

#### Product types

* Formation rules for product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \times B \; \mathrm{type}}$$

* Introduction rules for product types:

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B}{\Gamma \vdash (a, b):A \times B}$$

* Elimination rules for product types:

$$\frac{\Gamma \vdash z:A \times B}{\Gamma \vdash \pi_1(z):A} \qquad \frac{\Gamma \vdash z:A \times B}{\Gamma \vdash \pi_2(z):B}$$

* Computation rules for product types:

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B}{\Gamma \vdash \pi_1(a, b) \equiv_A a \; \mathrm{true}} \qquad \frac{\Gamma \vdash a:A \quad \Gamma \vdash b:B}{\Gamma \vdash \pi_2(a, b) \equiv_B b \; \mathrm{true}}$$

* Uniqueness rules for product types:

$$\frac{\Gamma \vdash z:A \times B}{\Gamma \vdash z \equiv_{A \times B} (\pi_1(z), \pi_2(z)) \; \mathrm{true}}$$

#### Dependent sum types

* Formation rules for dependent sum types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \sum_{x:A} B(x) \; \mathrm{type}}$$

* Introduction rules for dependent sum types:

$$\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:B[a/x]}{\Gamma \vdash (a, b):\sum_{x:A} B(x)}$$

* Elimination rules for dependent sum types:

$$\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash \pi_1(z):A} \qquad \frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash \pi_2(z):B(\pi_1(z))}$$

* Computation rules for dependent sum types:

$$\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \pi_1(a, b) \equiv_A a \; \mathrm{true}} \qquad \frac{\Gamma, x:A \vdash b:B \quad \Gamma \vdash a:A}{\Gamma \vdash \pi_2(a, b) \equiv_{B(\pi_1(a, b))} b \; \mathrm{true}}$$

* Uniqueness rules for dependent sum types:

$$\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash z \equiv_{\sum_{x:A} B(x)} (\pi_1(z), \pi_2(z)) \; \mathrm{true}}$$

#### Sum types

* Formation rules for sum types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A + B \; \mathrm{type}}$$

* Introduction rules for sum types:

$$\frac{\Gamma \vdash a:A}{\Gamma \vdash \mathrm{inl}(a):A + B} \qquad \frac{\Gamma \vdash b:B}{\Gamma \vdash \mathrm{inr}(b):A + B}$$

* Elimination rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B}{\Gamma \vdash \mathrm{ind}_{A + B}(z.C, c, d, e):C[e/z]}$$

* Computation rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash a:A}{\Gamma \vdash \mathrm{ind}_{A + B}(z.C, c, d, \mathrm{inl}(a)) \equiv_{C[\mathrm{inl}(a)/z]} c[a/x] \; \mathrm{true}}$$
$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash b:B}{\Gamma \vdash \mathrm{ind}_{A + B}(z.C, c, d, \mathrm{inr}(b)) \equiv_{C[\mathrm{inr}(b)/z]} d[b/y] \; \mathrm{true}}$$

* Uniqueness rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B \quad \Gamma, z:A + B \vdash u:C \quad \Gamma, a:A \vdash u[\mathrm{inl}(a)/z] \equiv_{C[\mathrm{inl}(a)/z]} c[a/x] \; \mathrm{true} \quad \Gamma, b:B \vdash u[\mathrm{inr}(b)/z] \equiv_{C[\mathrm{inr}(b)/z]} d[b/y] \; \mathrm{true}}{\Gamma \vdash u[e/z] \equiv_{C[e/z]} \mathrm{ind}_{A + B}(z.C, c[\mathrm{inl}(e)/x], d[\mathrm{inr}(e)/y], e) \; \mathrm{true}}$$

#### Empty type

* Formation rules for the empty type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{0} \; \mathrm{type}}$$

* Elimination rules for the empty type:

$$\frac{\Gamma, x:\mathbb{0} \vdash C \; \mathrm{type} \quad \Gamma \vdash p:\mathbb{0}}{\Gamma \vdash \mathrm{ind}_\mathbb{0}(x.C, p):C[p/x]}$$

* Uniqueness rules for the empty type:

$$\frac{\Gamma, x:\mathbb{0} \vdash C \; \mathrm{type} \quad \Gamma \vdash p:\mathbb{0} \quad \Gamma, x:\mathbb{0} \vdash u:C}{\Gamma \vdash u[p/x] \equiv_{C[p/x]} \mathrm{ind}_\mathbb{0}(x.C, p) \; \mathrm{true}}$$

#### Unit type

* Formation rules for the unit type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{1} \; \mathrm{type}}$$

* Introduction rules for the unit type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash *:\mathbb{1}}$$

* Elimination rules for the unit type:

$$\frac{\Gamma, x:\mathbb{1} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_*:C[*/x] \quad \Gamma \vdash p:\mathbb{1}}{\Gamma \vdash \mathrm{ind}_\mathbb{1}(x.C, p, c_*):C[p/x]}$$

* Computation rules for the unit type:

$$\frac{\Gamma, x:\mathbb{1} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_*:C[*/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{1}(x.C, *, c_*) \equiv_{C[*/x]} c_* \; \mathrm{true}}$$

* Uniqueness rules for the unit type:

$$\frac{\Gamma, x:\mathbb{1} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_*:C[*/x] \quad \Gamma \vdash p:\mathbb{1} \quad \Gamma, x:\mathbb{1} \vdash u:C \quad \Gamma \vdash u[*/x] \equiv_{C[*/x]} c_* \; \mathrm{true}}{\Gamma \vdash u[p/x] \equiv_{C[p/x]} \mathrm{ind}_\mathbb{1}(x.C, p, c_*) \; \mathrm{true}}$$

#### Natural numbers

* Formation rules for the natural numbers:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{N} \; \mathrm{type}}$$

* Introduction rules for the natural numbers:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{N}} \qquad \frac{\Gamma \vdash n:\mathbb{N}}{\Gamma \vdash s(n):\mathbb{N}}$$

* Elimination rules for the natural numbers:
$$\frac{\Gamma, x:\mathbb{N} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_0:C[0/x] \quad \Gamma, x:\mathbb{N}, c:C \vdash c_s:C[s(x)/x] \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C(n, c_0, c_s):C[n/x]}$$

* Computation rules for the natural numbers:
$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma, x:\mathbb{N}, c:C \vdash c_s:C[s(x)/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C(0, c_0, c_s) \equiv_{C[0/x]} c_0 \; \mathrm{true}}$$

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma, x:\mathbb{N}, c:C \vdash c_s:C[s(x)/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C(s(n), c_0, c_s) \equiv_{C[s(n)/x]} c_s(n, \mathrm{ind}_\mathbb{N}^C(n, c_0, c_s)) \; \mathrm{true}}$$

* Uniqueness rules for the natural numbers:
$$\frac{\Gamma, x:\mathbb{N} \vdash C \; \mathrm{type} \quad \Gamma \vdash c_0:C[0/x] \quad \Gamma, x:\mathbb{N}, c:C \vdash c_s:C[s(x)/x] \quad \Gamma \vdash n:\mathbb{N} \quad \Gamma, x:\mathbb{N} \vdash u:C \quad \Gamma \vdash i_0(u):u[0/x] =_{C[0/x]} c_0 \quad \Gamma, x:\mathbb{N} \vdash i_s(u):u[s(x)/x] =_{C[s(x)/x]} c_s[u/c]}{\Gamma \vdash u[n/x] =_{C[n/x]} \mathrm{ind}_\mathbb{N}^{C}(p, c_0, c_s) \; \mathrm{true}}$$

#### Identity types

There is another version of equality, called the [[identity type]] or **typal equality**, because this equality is a type. The identity type is only be used for equality between terms of a type, and unless one is using [[Russell universes]] where types are represented by terms of a [[Russell universe]], identity types cannot be used for equality of types.

* Formation rule for identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A, b:A \vdash a =_A b \; \mathrm{type}}$$

* Introduction rule for identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A \vdash \mathrm{refl}_A(a) : a =_A a}$$

* Elimination rule for identity types:

$$\frac{\Gamma, a:A, b:A, p:a =_A b \vdash C \mathrm{type} \quad \Gamma, c:A \vdash t:C[c/a, c/b, \mathrm{refl}_A(c)/p]}{\Gamma, a:A, b:A, p:a =_A b \vdash J(t, a, b, p):C}$$

* Computation rules for identity types:

$$\frac{\Gamma, a:A, b:A, p:a =_A b \vdash C \; \mathrm{type} \quad \Gamma, c:A \vdash t:C[c/a, c/b, \mathrm{refl}_A(c)/p]}{\Gamma, c:A \vdash J(t, c, c, \mathrm{refl}(c)) \equiv_{C[c/a, c/b, \mathrm{refl}_A(c)/p]} t \; \mathrm{true}}$$

* Optional uniqueness rules for identity types:

...

#### Intensional and extensional type theories

In Martin-Löf dependent type theory, one makes a distinction between intensional and extensional type theories. A Martin-Löf dependent type theory is an [[extensional type theory]] if there is an equality reflection rule where from typal equality of terms, one could derive definitional equality of terms:

$$\frac{\Gamma, a:A, b:A \vdash p:a =_A b}{\Gamma, a:A, b:A \vdash a \equiv_A b \; \mathrm{true}}$$

The type theory is an [[intensional type theory]] if it doesn't have the equality reflection rule.

#### Propositional reflection and predicate logic

One could add an additional set of rules in this second presentation of Martin-Löf dependent type theory which isn't available in the first presentation: propositional reflections. Unlike the first presentation, in this presentation, there are two notions of propositions: the external propositions and the internal propositions. The external propositions are the judgmental propositions in the logic layer of Martin-Löf dependent type theory, and correspond to the external logic. On the other hand, the internal propositions are [[types]] equipped with a witness of the [[isProp]] [[modality]] and correspond to the [[internal logic]]. They are also called [[subsingletons]] or [[h-propositions]]. We could make the internal logic imply the external logic with the following proposition reflection rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \phi_A \; \mathrm{prop}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash a:A}{\Gamma \vdash \phi_A \; \mathrm{true}}$$

and similar rules for [[predicates]]:

$$\frac{\Gamma, x:A, \Delta \vdash B \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash p:\mathrm{isProp}(B)}{\Gamma, x:A, \Delta \vdash \phi_B \; \mathrm{prop}} \qquad \frac{\Gamma, x:A, \Delta \vdash B \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash p:\mathrm{isProp}(B) \quad \Gamma, x:A, \Delta \vdash b:B}{\Gamma, x:A, \Delta \vdash \phi_B \; \mathrm{true}}$$

If the [[identity types]] of Martin-Löf dependent type theory have a uniqueness rule such as the [[K rule]] or [[uniqueness of identity proofs]], then propositional reflection implies that the type theory is an [[extensional type theory]].

In addition, since the [[product type]] and [[function type]] preserve types being a subsingleton, the dependent product of a subsingleton-valued type family, and the [[unit type]] and [[empty type]] are always subsingletons, one could also add the following propositional reflection rules to the theory, defining [[conjunction]], [[implication]], [[true]], [[false]], and bounded [[universal quantification]]:

$$\frac{\Gamma \vdash A \times B \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:\mathrm{isProp}(B)}{\Gamma \vdash \phi_A \wedge \phi_B \; \mathrm{prop}}$$
$$\frac{\Gamma \vdash A \to B \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:\mathrm{isProp}(B)}{\Gamma \vdash \phi_A \implies \phi_B \; \mathrm{prop}}$$
$$\frac{\Gamma \vdash \mathbb{0} \; \mathrm{type}}{\Gamma \vdash \bot \; \mathrm{prop}} \qquad \frac{\Gamma \vdash \mathbb{1} \; \mathrm{type}}{\Gamma \vdash \top \; \mathrm{prop}}$$
$$\frac{\Gamma, \Delta \vdash \prod_{x:A} B(x) \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash q:\mathrm{isProp}(B(x))}{\Gamma, \Delta \vdash \forall x:A.\phi_B(x) \; \mathrm{prop}}$$

Adding [[propositional truncations]] to Martin-Löf dependent type theory allows for the full [[intuitionistic logic]] to be defined, as [[disjunctions]] could be defined from the propositional truncation of the [[sum type]], and the [[existential quantifier]] could be defined from the propositional truncation of the [[dependent sum type]]:

$$\frac{\Gamma \vdash [A + B] \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma\vdash q:\mathrm{isProp}(B)}{\Gamma \vdash \phi_A \vee \phi_B \; \mathrm{prop}}$$
$$\frac{\Gamma, \Delta \vdash \left[\sum_{x:A} B(x)\right] \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash q:\mathrm{isProp}(B(x))}{\Gamma, \Delta \vdash \exists x:A.\phi_B(x) \; \mathrm{prop}}$$

The rules above reflect the [[propositions as some types]] philosophy in [[type theory]].

#### Bounded separation

A predicate on a type $A$ is a proposition $P \; \mathrm{prop}$ in the context of the free variable $x:A$. Bounded separation states that given a predicate $P$ on a set $A$, one could construct the set $\vert P \vert$ with an injection $i:\vert P \vert \hookrightarrow A$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash P \; \mathrm{prop}}{\Gamma \vdash \vert P \vert \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash P \; \mathrm{prop}}{\Gamma \vdash i:\vert P \vert \to A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash P \; \mathrm{prop}}{\Gamma, x:A, y:A \vdash \varphi_{x =_A y} \iff \varphi{i(x) =_B i(y)} \; \mathrm{true}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x \in A \vdash P \; \mathrm{prop}}{\Gamma, x \in A \vdash P(x) \iff \varphi_{\sum_{y:\vert P \vert} i(y) = x} \; \mathrm{true}}$$

#### Propositions as types

An alternative to the [[propositions as some types]] philosophy is the [[propositions as types]] philosophy, where instead of reflecting the [[subsingletons]] to propositions and the [[singletons]] to true propositions, we reflect all types to propositions, and the [[pointed types]] to true propositions. The rules are given as follows:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \phi_A \; \mathrm{prop}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash \phi_A \; \mathrm{true}}$$

and similar rules for [[predicates]]:

$$\frac{\Gamma, x:A, \Delta \vdash B \; \mathrm{type}}{\Gamma, x:A, \Delta \vdash \phi_B \; \mathrm{prop}} \qquad \frac{\Gamma, x:A, \Delta \vdash B \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash b:B}{\Gamma, x:A, \Delta \vdash \phi_B \; \mathrm{true}}$$

There is no need for [[propositional truncations]] in this interpretation: the full [[intuitionistic logic]] results from reflecting all type formers except for the [[natural numbers type]] and [[identity types]]:

$$\frac{\Gamma \vdash A + B \; \mathrm{type}}{\Gamma \vdash \phi_A \vee \phi_B \; \mathrm{prop}}$$
$$\frac{\Gamma \vdash A \times B \; \mathrm{type}}{\Gamma \vdash \phi_A \wedge \phi_B \; \mathrm{prop}}$$
$$\frac{\Gamma \vdash A \to B \; \mathrm{type}}{\Gamma \vdash \phi_A \implies \phi_B \; \mathrm{prop}}$$
$$\frac{\Gamma \vdash \mathbb{0} \; \mathrm{type}}{\Gamma \vdash \bot \; \mathrm{prop}} \qquad \frac{\Gamma \vdash \mathbb{1} \; \mathrm{type}}{\Gamma \vdash \top \; \mathrm{prop}}$$
$$\frac{\Gamma \vdash \sum_{x:A} B(x) \; \mathrm{type}}{\Gamma \vdash \exists x:A.\phi_B(x) \; \mathrm{prop}}$$
$$\frac{\Gamma \vdash \prod_{x:A} B(x) \; \mathrm{type}}{\Gamma \vdash \forall x:A.\phi_B(x) \; \mathrm{prop}}$$

## Properties

### 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]] ([Warren](#Warren)). For this reason one speaks of _[[homotopy type theory]]_.

For a more detailed discussion of these matters see at _[[relation between type theory and category theory]]_.

### Axiom of choice
 {#AxiomOfChoice}

In dependent type theory we can verify the following "logical form of the [[axiom of choice]]" ([Bell](#Bell), [Tait](#Tait)), see also ([Rijke, section 2.5.1](#Rijke)).

+-- {: .num_theorem }
###### Theorem
**(ACL)**

$$
  x : A, y : B(x) \vdash C(x, y) : Type
  \;
  \vdash
  \;
  acl
  :
  (\forall x : A) (\exists y : B(x)) C(x, y) \to (\exists f : (\Pi x : A)B(x) )(\forall x : A) C(x, \mathrm{apply}(f, x))
  \,.
$$

=--

One should note carefully that this "is" only "the axiom of choice" under the above propositions-as-types interpretation of the quantifiers $\forall$ and $\exists$.

+-- {: .num_remark}
###### Remark

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.

=--

+-- {: .num_remark}
###### Remark

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.

=--

## Related concepts

* [[initiality conjecture]]

[[!include notions of type]]

## References
 {#References}

See also the references at _[[type theory]]_ and at *[[dependent type theory]]*.

The original articles:

* [[Per Martin-Löf]], *A Theory of Types*, unpublished note (1971) [[pdf](https://raw.githubusercontent.com/michaelt/martin-lof/master/pdfs/martin-loef1971%20-%20A%20Theory%20of%20Types.pdf), [[MartinLoef1971-ATheoryOfTypes.pdf:file]]]

* {#MartinLof75} [[Per Martin-Löf]], _An intuitionistic theory of types: predicative part_, in: H. E. Rose, J. C. Shepherdson (eds.), *Logic Colloquium '73, Proceedings of the Logic Colloquium*, Studies in Logic and the Foundations of Mathematics **80**, Elsevier (1975) 73-118 [doi:10.1016/S0049-237X(08)71945-1, [CiteSeer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926)]

* [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]), _Intuitionistic type theory_, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) ([pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]])

As a [[programming language]]:

* [[Per Martin-Löf]], *Constructive Mathematics and Computer Programming*, in: *Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science (1979)*, Studies in Logic and the Foundations of Mathematics **104** (1982) 153-175 [doi:10.1016/S0049-237X(09)70189-2, [ISBN:978-0-444-85423-0](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/104/suppl/C)]

* Bengt Nordström, Kent Petersson, Jan M. Smith, *Programming in Martin-Löf's Type Theory*, Oxford University Press 1990 ([webpage](http://www.cse.chalmers.se/research/group/logic/book/), [pdf](http://www.cse.chalmers.se/research/group/logic/book/book.pdf))

Further discussion:

* {#Hofmann95} [[Martin Hofmann]], *Syntax and semantics of dependent types*, Chapter 2 in: _Extensional concepts in intensional type theory_, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [[ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]], [doi:10.1007/978-1-4471-0963-1](https://doi.org/10.1007/978-1-4471-0963-1)]

also published as:

* [[Martin Hofmann]], *Syntax and semantics of dependent types*, in *Semantics and logics of computation*, Publ. Newton Inst. **14**, Cambridge Univ. Press (1997) 79-130 [[doi:10.1017/CBO9780511526619.004](https://doi.org/10.1017/CBO9780511526619.004)]

Textbook accounts on general [[dependent type theory]]:

* {#Thompson91} [[Simon Thompson]], §6.3 in: *[[Type Theory and Functional Programming]]*, Addison-Wesley (1991) [ISBN:0-201-41667-0, [webpage](http://www.cs.kent.ac.uk/people/staff/sjt/TTFP), [pdf](http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf)]

* {#Jacobs98} [[Bart Jacobs]], Chapter 10 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

  > (emphasis on the [[categorical model of dependent types]])

* {#UFP13} [[Univalent Foundations Project]], §A of *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

  > (in the context of [[homotopy type theory]])

* {#Rijke18} [[Egbert Rijke]], *Dependent type theory* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/dtt.pdf)], Lecture 1 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]

  > (in the context of [[homotopy type theory]])

A philosophical examination:

* [[Göran Sundholm]], _Proof Theory and Meaning_ (1986), ([pdf](https://openaccess.leidenuniv.nl/bitstream/handle/1887/11978/9_05?sequence=1))

An introduction and survey (with an eye towards [[homotopy type theory]]):

* {#Warren} [[Michael Warren]], chapter 1 of: _Homotopy theoretic aspects of constructive type theory_, PhD thesis (2008) ([pdf](http://www.andrew.cmu.edu/user/awodey/students/warren.pdf))

as well as

* {#Rijke} [[Egbert Rijke]], _Homotopy type theory_ (2012) ([pdf](http://hottheory.files.wordpress.com/2012/08/hott2.pdf))

A discussion of ML dependent type theory as the [[internal language]] of [[locally cartesian closed categories]] is in

* [[R. A. G. Seely]], _Locally cartesian closed categories and type theory_, Math. Proc. Camb. Phil. Soc. (1984) 95 ([pdf](http://www.math.mcgill.ca/rags/LCCC/LCCC.pdf))

Discussion of the logical [[axiom of choice]] in dependent type theory is in

* {#Bell} John Bell, _The Axiom of choice in type theory_, Stanford Encyclopedia of philosophy, 2008 ([web](http://plato.stanford.edu/entries/axiom-choice/choice-and-type-theory.html))

* {#Tait} Tait (1994)

[[!redirects Martin-Löf type theory]]
[[!redirects Martin-Löf type theories]]
[[!redirects Martin-Löf dependent type theories]]

[[!redirects Martin-Löf's dependent type theory]]
[[!redirects Martin-Löf's dependent type theories]]

[[!redirects Martin-Lof type theory]]
[[!redirects Martin-Lof type theories]]
[[!redirects Martin-Lof dependent type theory]]
[[!redirects Martin-Löf dependent type theory]]

[[!redirects Martin-Loef type theory]]
[[!redirects Martin-Loef type theories]]
[[!redirects Martin-Loef dependent type theory]]
[[!redirects Martin-Loef dependent type theories]]

[[!redirects constructive type theory]]

[[!redirects intuitionistic type theory]]
[[!redirects intuitionistic type theories]]

[[!redirects intuitionistic dependent type theory]]
[[!redirects intuitionistic dependent type theories]]

[[!redirects MLTT]]


Martina Rovelli

Martina Rovelli is an Assistant Professor at University of Massachusetts Amherst. Her research interests are in Algebraic topology, Homotopy theory and Higher category theory.

* [webpage](https://people.math.umass.edu/~rovelli/)

## Selected Publications

On [[weighted limit|weighted]] [[(infinity,1)-limits|$(\infty,1)$-limits]]:

* [[Martina Rovelli]], _Weighted limits in an (∞,1)-category_ (2019) [[arxiv:1902.00805](https://arxiv.org/abs/1902.00805)]

On the [[adjunction unit]] of [[total décalage]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *The unit of the total décalage adjunction*, Journal of Homotopy and Related Structures **15** (2020) 333–349 [[doi.org/10.1007/s40062-020-00257-1](https://doi.org/10.1007/s40062-020-00257-1)]

A [[model category]] structure on [[stratified simplicial sets]] modelling [[(infinity,n)-categories|$(\infty,n)$-categories]] in the guise of [[n-complicial sets|$n$-complicial sets]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces*, Algebr. Geom. Topol. **20** (2020) 1543-1600 [[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), [doi:10.2140/agt.2020.20.1543](https://doi.org/10.2140/agt.2020.20.1543)]

A [[Quillen adjunction]] relating [[n-complicial sets|$n$-complicial sets]] to [[n-fold complete Segal spaces|$n$-fold complete Segal spaces]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories*, Advances in Mathematics **421** (2023) 108980 [[doi:10.1016/j.aim.2023.108980](https://doi.org/10.1016/j.aim.2023.108980)]

Discussion of straightening and unstraightening for [[(∞,n)-categories]]:

* [[Lyne Moser]], [[Nima Rasekh]], [[Martina Rovelli]], _An (∞,n)-categorical straightening-unstraightening construction_ ([arXiv:2307.07259](https://arxiv.org/abs/2307.07259)).

category: people


Martindale localization

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## Idea

A *Martindale quotient* or *Martindale localization* is a kind of symmetric [[quotient]] for some types of [[rings]] and [[algebras]]. The principal construction is for [[prime rings]].

## Related concepts

* [[localization of a ring]]

* [[Cohn localization]]

## References

* J. A. Anquela, K. McCrimmon, _Martindale Quotients of Jordan Algebras_, [pdf](http://molle.fernuni-hagen.de/~loos/jordan/archive/martindaleq/martindaleq.pdf)

* J. D. Rosen, M. P. Rosen, _The martindale ring of quotients of a skew polynomial ring of automorphism type_, Communications in Algebra
__21__:11, 4051-4063, 1993 [doi](http://dx.doi.org/10.1080/00927879308824783)

* Springer [eom], *[Martindale ring of quotients](http://www.encyclopediaofmath.org/index.php/Martindale_ring_of_quotients)*

[[!redirects Martindale quotient]]


Martindale ring of quotients

+-- {: .rightHandSide}
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## Related concepts

* [[prime ring]]

## References

* [[eom]], _[Martindale ring of quotients](https://www.encyclopediaofmath.org/index.php/Martindale_ring_of_quotients)_

[[!redirects Martindale rings of quotients]]

[[!redirects Amitsur-Martindale ring of quotients]]
[[!redirects Amitsur-Martindale rings of quotients]]


Martintxo Saralegi-Aranguren

* [webpage](http://perso.numericable.fr/saralegi2/MS/ms.html)

## related $n$Lab entries

* [[circle action]]

category: people

[[!redirects Martin Saralegi]]


Martinus Veltman

* [Wikipedia entry](https://en.wikipedia.org/wiki/Martinus_J._G._Veltman)

## Selected writings

Proving the non-[[renormalizable interaction|renormalizability]] of Einstein-[[gravity]] ([[general relativity]]):

* {#tHooftVelrman74} [[Gerard 't Hooft]], [[Martinus Veltman]], _One loop divergencies in the theory of gravitation_, Ann. Inst. Poincar é 20 (1974) 69 ([numdam:AIHPA_1974__20_1_69_0](AIHPA_1974__20_1_69_0))

On [[quantum gravity]]:

* [[Martinus Veltman]], _Quantum theory of Gravitation_, In Les Houches 1975, Proceedings, Methods In Field Theory, (Amsterdam 1976) 265-327 ([[VeltmanQuantumnGravity76.pdf:file]], [spire](http://inspirehep.net/record/105735?ln=en))

## related $n$Lab entries

* [[gravity]]

* [[perturbative quantum gravity]]

category: people


Martti Karvonen

* [Personal webpage](https://marttikarvonen.cs.ucl.ac.uk/)

## Selected writings

On [[dagger categories]]:

* {#Karvonen18} [[Martti Karvonen]], *The Way of the Dagger*, Edinburgh (2018) [[arXiv:1904.10805](https://arxiv.org/abs/1904.10805), [pdf](https://mysite.science.uottawa.ca/mkarvone/papers/Karvonen%20-%20Thesis.pdf)]

and on [[limits]] in [[dagger categories]]:

* [[Chris Heunen]], [[Martti Karvonen]], *Limits in dagger categories*, Theory and Applications of Categories **34** 18 (2019) 468-513 [[arXiv:1803.06651](https://arxiv.org/abs/1803.06651), [34-18](http://www.tac.mta.ca/tac/volumes/34/18/34-18abs.html)]

On [[monads]] on [[dagger categories]] with an eye towards [[quantum measurement]] expressed in [[finite quantum mechanics in terms of dagger-compact categories]]:

* [[Chris Heunen]], [[Martti Karvonen]], *Monads on dagger categories*, Theory and Applications of Categories **31** 35 (2016) 1016-1043 [[arXiv:1602.04324](https://arxiv.org/abs/1602.04324), [TAC:31-35](http://www.tac.mta.ca/tac/volumes/31/35/31-35abs.html)]

On [[biproducts]]:

* {#Karvonen20} [[Martti Karvonen]], _Biproducts without pointedness_,  Cahiers de topologie et géométrie différentielle catégoriques **61** 3 (2020) 229–238 $[$[arXiv:1801.06488](https://arxiv.org/abs/1801.06488), [journal pdf](http://cahierstgdc.com/wp-content/uploads/2020/07/KARVONEN-LXI-3.pdf)$]$

## Related pages

* [[biproduct]]

* [[semiadditive dagger category]]

category: people


Martín Escardó

* [webpage](http://www.cs.bham.ac.uk/~mhe/)

## Selected writings

On [[synthetic topology]]:

* [[Martín Escardó]], _Synthetic topology of data types and classical spaces_  [[pdf](http://www.cs.bham.ac.uk/~mhe/papers/barbados.pdf)]

* [[Martín Escardó]], {#EscardoPopl2012} _The topology of Seemingly impossible functional programs_ (2012) [[pdf](http://www.cs.bham.ac.uk/~mhe/.talks/popl2012/escardo-popl2012.pdf)]

On [[exact real computer arithmetic]]:

* Dave Plume (supervised by [[Martín Escardó]], [[Alex Simpson]]): *A Calculator for Exact Real Number Computation*, University of Edinburgh  (1998) [[web](https://www.dcs.ed.ac.uk/home/mhe/plume/)]

On [[differential calculus]] via [[coinduction]]:

* [[Martín Escardó]], [[Duško Pavlović]], _Calculus in coinductive form_ (1998) ([pdf](http://www.isg.rhul.ac.uk/dusko/papers/1998-lapl-LICS.pdf))

On [[cartesian closed category|cartesian closed]] [[convenient categories of topological spaces]], such as [[compactly generated topological spaces]]:

* {#EscardoLawsonSimpson04} [[Martín Escardó]], [[Jimmie Lawson]], [[Alex Simpson]], *Comparing Cartesian closed categories of (core) compactly generated spaces*, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 ([doi:10.1016/j.topol.2004.02.011](https://doi.org/10.1016/j.topol.2004.02.011))

On the [[exponential law for spaces]]:

* [[Martín Escardó]], [[Reinhold Heckmann]], *Topologies of spaces of continuous functions* (2001) $[$[pdf](http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf), [[EscardoHeckmann-TopologiesOnFunctionSpaces.pdf:file]]$]$

On [[Hedberg's theorem]]:

* {#KECA} [[Nicolai Kraus]], [[Martin Escardo]], [[Thierry Coquand]] , [[Thorsten Altenkirch]], _Generalizations of Hedberg's theorem_, in M. Hasegawa (ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. ([pdf](http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf))

On [[compactness and stable closure]]:

* [[Martín Escardó]], Intersections of compactly many open sets are open, 2009 [(pdf)](http://www.cs.bham.ac.uk/~mhe/papers/compactness-submitted.pdf)

On [[weakly constant functions]] and [[propositional truncation]] in [[homotopy type theory]]:

* [[Nicolai Kraus]], [[Martín Escardó]], [[Thierry Coquand]], [[Thorsten Altenkirch]], *Notions of Anonymous Existence in Martin-Löf Type Theory*,  Logical Methods in Computer Science **13** 1 [doi:10.23638/LMCS-13(1:15)2017, [arXiv:1610.03346](https://arxiv.org/abs/1610.03346)]

On [[injective objects]] in [[homotopy type theory]]:

* {#Escardo19} [[Martín Escardó]], _Injectives types in univalent mathematics_ ([arXiv:1903.01211](https://arxiv.org/abs/1903.01211))

On [[homotopy type theory]] and [[univalent foundations of mathematics]] in/with [[Agda]]:

* [[Martín Hötzel Escardó]], *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580), [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/)]

On the [[selection monad]]:

* [[Martín Escardó]] and Paulo Oliva, _Selection Functions, Bar Recursion, and Backward Induction_, ([pdf](https://www.cs.bham.ac.uk/~mhe/papers/selection-escardo-oliva.pdf))

* [[Martín Escardó]] and Paulo Oliva, _What Sequential Games, the Tychonoff Theorem and the Double-Negation Shift have in Common_, ([pdf](https://www.cs.bham.ac.uk/~mhe/papers/msfp2010/Escardo-Oliva-MSFP2010.pdf))

*  {#Escardó-Oliva2012} [[Martín Escardó]] and Paulo Oliva, _The Peirce translation_, Annals of Pure and Applied Logic, 163(6):681–692, 2012, ([pdf](https://www.cs.bham.ac.uk/~mhe/papers/peirce.pdf)).

On [[domain theory]] in [[homotopy type theory]]/[[univalent foundations]] ([[mathematics presented in HoTT]]):

* [[Tom de Jong]], [[Martín Hötzel Escardó]], _Domain Theory in Constructive and Predicative Univalent Foundations_, in: _29th EACSL Annual Conference on Computer Science Logic_, [CSL 2021](https://csl2021.fmf.uni-lj.si/), LIPIcs proceedings 183, 2021 ([doi:10.4230/LIPIcs.CSL.2021.28](https://doi.org/10.4230/LIPIcs.CSL.2021.28), [arXiv:2008.01422](https://arxiv.org/abs/2008.01422))

category: people

[[!redirects Martin Escardo]]
[[!redirects Martín Hötzel Escardó]]


Maru Sarazola

* [personal page](https://sites.google.com/view/msarazola/home)

## Selected writings

On [[locally finitely presentable categories]]:

* [[Maru Sarazola]], *An introduction to locally finitely presentable categories* (2017) [[pdf](https://pi.math.cornell.edu/~maru/documents/locally_finitely_presentable_cats.pdf), [[Sarazola_LocallyPresentableCategories.pdf:file]]]

category: people


Marwa Mannaï

member of [[CQTS]]

## Selected writings

Models for [[Chern insulators]] akin to the [[Haldane model]]:

* [[Marwa Mannaï]], [[Jean-Noël Fuchs]], [[Frédéric Piéchon]], [[Sonia Haddad]], *Stacking-induced Chern insulator* [[arXiv:2208.02491](https://arxiv.org/abs/2208.02491)]

On the behaviour of [[topological insulators]]/[[semi-metals]] ([[Haldane model]]) under external [[strain]]:

* {#MannaïHaddad20} [[Marwa Mannaï]], [[Sonia Haddad]], *Strain tuned topology in the Haldane and the modified Haldane models*., J of Physics: Condens. Matter **32** 225501 (2020) $[$[arXiv:1907.11213](https://arxiv.org/abs/1907.11213), [doi:10.1088/1361-648X/ab73a1](https://doi.org/10.1088/1361-648X/ab73a1)$]$

* [[Marwa Mannaï]], [[Sonia Haddad]], *Twistronics versus straintronics in twisted bilayers of graphene and transition metal dichalcogenides*, Phys. Rev. B **103** 201112 (2021) $[$[arXiv:2011.08818](https://arxiv.org/abs/2011.08818), [doi:10.1103/PhysRevB.103.L121112](https://doi.org/10.1103/PhysRevB.103.L121112)$]$

category: people

[[!redirects Marwa Mannai]]


Maryam Mirzakhani

* [Wikipedia entry](http://en.wikipedia.org/wiki/Maryam_Mirzakhani)

## related $n$Lab entries

* [[prime geodesic theorem]]

* [[Witten conjecture]]

category: people


Maryna Viazovska

* [Wikipedia entry](https://en.wikipedia.org/wiki/Maryna_Viazovska)

* awarded a [Field medal 2022](https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2022) for, mostly, the work of [Viazovska 2016](#Viazovska16).

## Selected writings

On the solution to the [[sphere packing problem]] in [[dimension]] 8 via the [[E8 lattice]]:

* {#Viazovska16} [[Maryna Viazovska]], _The sphere packing problem in dimension 8_, Annals of Mathematics, **185** 3 (2017) 991-1015 [[arXiv:1603.04246](https://arxiv.org/abs/1603.04246), [doi:10.4007/annals.2017.185.3.7](https://doi.org/10.4007/annals.2017.185.3.7)]

and in [[dimension]] 24 via the [[Leech lattice]]

* {#CKMRV17} Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, [[Maryna Viazovska]], _The sphere packing problem in dimension 24_, Annals of Mathematics **185** (2017) 1017-1033 [[arXiv:1603.06518](https://arxiv.org/abs/1603.06518), [doi:10.4007/annals.2017.185.3.8](https://doi.org/10.4007/annals.2017.185.3.8)]

category: people


Marzia Bordone

* [personal page](https://www.marziabordone.com/)

## Selected writings

On [[QED]]-[[radiative corrections]] to branching ratios in the discussion of [[flavour anomalies]]:

* {#BordoneIsidoriPattori16} [[Marzia Bordone]], [[Gino Isidori]], Andrea Pattori, *On the Standard Model predictions for $R_K$ and $R_{K^\ast}$*, Eur. Phys. J. C 76, 440 (2016). ([arXiv:1605.07633](https://arxiv.org/abs/1605.07633), [doi:10.1140/epjc/s10052-016-4274-7](https://doi.org/10.1140/epjc/s10052-016-4274-7))

category: people


Marzieh Bayeh

* [Institute page](https://www.mathstat.dal.ca/bayeh/)

## Selected writings

On left-[[transferred model structures]] and [[model structures on functors]]:

* [[Marzieh Bayeh]], [[Kathryn Hess]], [[Varvara Karpova]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *Left-induced model structures and diagram categories*, in: *Women in Topology: Collaborations in Homotopy Theory*, Contemporary Mathematics **641** American Mathematical Society (2015) [[arXiv:1401.3651](http://arxiv.org/abs/1401.3651), [ISBN:978-1-4704-2495-4](https://bookstore.ams.org/conm-641)]

category: people


Masahico Saito

[[!redirects M. Saito]]

* [Homepage](http://shell.cas.usf.edu/~saito/)

##Publications

* [[Mikhail Kapranov]] and [[Masahico Saito]], 1999, _Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions_, in _Higher homotopy structure in topology and mathematical physics_ (Poughkeepsie, N.Y. 1996), volume 227 of Contemporary Mathematics, 191 – 225, AMS.

* [[Scott Carter]], Seiichi Kamada, Masahico Saito, _Surfaces in 4-Space_, 2004, Springer.

[[!redirects M. Saito]]

category: people


Masahiro Sugawara

## Selected writings

On [[H-spaces]]:

* {#Sugawara60} [[Masahiro Sugawara]], *On the homotopy-commutativity of groups and loop spaces*, Mem. College Sci. Univ. Kyoto Ser. A Math. 33(2): 257-269 (1960) ([DOI:10.1215/kjm/1250775911](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-33/issue-2/On-the-homotopy-commutativity-of-groups-and-loop-spaces/10.1215/kjm/1250775911.full))

On detection of [[weak homotopy equivalences]] and [[equivariant weak homotopy equivalences]] in terms of free homotopy sets:

* [[Takao Matumoto]], [[Norihiko Minami]], [[Masahiro Sugawara]], *On the set of free homotopy classes and Brown's construction*, Hiroshima Math. J. 14(2): 359-369 (1984) ([doi:10.32917/hmj/1206133043](https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-14/issue-2/On-the-set-of-free-homotopy-classes-and-Browns-construction/10.32917/hmj/1206133043.full))

## Related $n$Lab entries

* [[H-space]]

* [[strong homotopy map]]


Masahito Hasegawa

* [webpage](http://www.kurims.kyoto-u.ac.jp/~hassei/)

## related $n$Lab entries

* [[traced monoidal category]]

category: people

[[!redirects Hassei]]


Masahito Hayashi

* [research group page](https://mhayashi.info/)

* [GoogleScholar page](https://scholar.google.com/citations?user=GGHbquQAAAAJ&hl=en)

## Selected writings

On [[quantum information theory]]:

* [[Masahito Hayashi]], _Quantum information theory - mathematical foundation_, Graduate Texts in Physics, Springer (2017) [[doi:10.1007/978-3-662-49725-8](https://doi.org/10.1007/978-3-662-49725-8)]

category: people


Masahito Yamazaki

* [webpage](http://member.ipmu.jp/masahito.yamazaki/index.shtml)

## Selected writings

Introducing the [[3d-3d correspondence]]:

* [[Yuji Terashima]], [[Masahito Yamazaki]], _$SL(2,\mathbb{R})$ Chern-Simons, Liouville, and Gauge Theory on Duality Walls_, JHEP 1108:135, 2011 ([arXiv:1103.5748](https://arxiv.org/abs/1103.5748))

* [[Yuji Terashima]], [[Masahito Yamazaki]], _Semiclassical Analysis of the 3d/3d Relation_, Phys.Rev.D88:026011, 2013 ([arXiv:1106.3066](https://arxiv.org/abs/1106.3066))

On [[semi-holomorphic 4d Chern-Simons theory]] and relation to [[integrable systems]]:

* {#CostelloWittenYamazaki17} [[Kevin Costello]], [[Edward Witten]], [[Masahito Yamazaki]], _Gauge Theory and Integrability, I_, ICCM Not. 6, 46-119 (2018) ([arXiv:1709.09993](https://arxiv.org/abs/1709.09993))

* {#CostelloWittenYamazaki17} [[Kevin Costello]], [[Edward Witten]], [[Masahito Yamazaki]], _Gauge Theory and Integrability, II_, ICCM Not. 6, 120-146 (2018) ([arXiv:1802.01579](https://arxiv.org/abs/1802.01579))

* [[Meer Ashwinkumar]], Jun-ichi Sakamoto, [[Masahito Yamazaki]], *Dualities and Discretizations of Integrable Quantum Field Theories from 4d Chern-Simons Theory* [[arXiv:2309.14412](https://arxiv.org/abs/2309.14412)]

## related $n$Lab entries

* [[semi-holomorphic 4d Chern-Simons theory]]

category: people


Masaki Kashiwara

[[Masaki Kashiwara]] is a Japanese mathematician. He made substantial contributions to the theory of [[pseudodifferential operator]]s, [[hyperfunction]]s and [[microlocal analysis]], study of [[D-module]]s, holonomic systems, Riemann-Hilbert problem, categories of [[constructible sheaf|constructible sheaves]] and [[representation theory]]. He invented [[crystal base|crystal bases]] and crystals in Lie and [[quantum group]] theory.

* [homepage](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara), [list of publicatons](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/list.html)

## Selected writings

On [[microlocalization]] of [[abelian sheaves]] and [[Grothendieck's yoga of six operations]]:

* [[Masaki Kashiwara]], [[Pierre Schapira]], _Sheaves on manifolds_, Grundlehren **292**, Springer (1990) [[doi:10.1007/978-3-662-02661-8](https://doi.org/10.1007/978-3-662-02661-8)]

On [[category theory]] and [[sheaf and topos theory]]:

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)

See also:

* P. Shapira, _Masaki Kashiwara and algebraic analysis_, Adv. Studies in Pure Analysis, [arxiv/0810.4875](http://arxiv.org/abs/0810.4875)

* Stephane Guillermou, Masaki Kashiwara, Pierre Schapira, _Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems_, [arxiv/1005.1517](http://arxiv.org/abs/1005.1517)

* M. Kashiwara, W.Schmid, _Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups_, Lie Theory and Geometry, in Honor of Bertram Kostant, Progress in Mathematics, Birkhäuser, 1994, pp. 457–488

* M. Kashiwara, _D-modules and representation theory of Lie groups_,
Annales de l'institut Fourier, 43 no. 5 (1993), p. 1597-1618, [article](http://aif.cedram.org/item?id=AIF_1993__43_5_1597_0), [MR95b:22033](http://www.ams.org/mathscinet-getitem?mr=95b:22033)

* M. Kashiwara, T. Kawai, T. Kimura, _Foundations of algebraic analysis_, Princeton Univ. Press 1986

* M. Kashiwara, _D-modules and microlocal analysis_,  Transl. of math. monographs __217__, Amer. Math. Soc. 2000
[[!redirects M. Kashiwara]]
[[!redirects Kashiwara]]

category:people


Masaki Oshikawa

* [Research group page](https://oshikawa.issp.u-tokyo.ac.jp/)

## Selected writings

On [[symmetry-protected topological order]]:

* {#PBTO09} [[Frank Pollmann]], [[Erez Berg]], [[Ari M. Turner]], [[Masaki Oshikawa]], *Symmetry protection of topological order in one-dimensional quantum spin systems*, Phys. Rev. B **85** 075125 (2012) $[$[arXiv:0909.4059](https://arxiv.org/abs/0909.4059), [doi:10.1103/PhysRevB.85.075125](https://doi.org/10.1103/PhysRevB.85.075125)$]$

On characterizing [[anyon]] [[braiding]] / [[modular transformations]] on [[topological order|topologically ordered]] [[ground states]] by analysis of ([[topological entanglement entropy|topological]]) [[entanglement entropy]] of subregions:

* [[Yi Zhang]], [[Tarun Grover]], [[Ari M. Turner]], [[Masaki Oshikawa]], [[Ashvin Vishwanath]], *Quasiparticle statistics and braiding from ground-state entanglement*, Phys. Rev. B **85** (2012) 235151 $[$[doi:10.1103/PhysRevB.85.235151](https://doi.org/10.1103/PhysRevB.85.235151)$]$

category: people


Masaki Shigemori

* [personal page](http://www2.yukawa.kyoto-u.ac.jp/~masaki.shigemori/)

## Selected writings

On [[exotic branes]] (in [[string theory]]) as [[defect branes]] with [[U-duality]]-group [[monodromy]]:

* [[Jan de Boer]], [[Masaki Shigemori]], _Exotic Branes in String Theory_, Physics Reports **532** (2013) 65-118 ([arXiv:1209.6056](https://arxiv.org/abs/1209.6056), [doi:j.physrep.2013.07.003](https://doi.org/10.1016/j.physrep.2013.07.003))

On [[defect branes]] in [[D=5 supergravity]]:

* Minkyu Park, [[Masaki Shigemori]], _Codimension-2 Solutions in Five-Dimensional Supergravity_, JHEP 1510 (2015) 011 ([arXiv:1505.05169](http://arxiv.org/abs/1505.05169))

* [[Masaki Shigemori]], *Interpolating between multi-center microstate geometries*, JHEP 09 (2021) 010 ([arXiv:2105.11639](https://arxiv.org/abs/2105.11639))

On [[AdS-CFT duality|AdS-CFT dual]] [[type II supergravity]]-solutions corresponding to [[D1-D5-P bound states]] ("[[superstrata]]"):

* [[Iosif Bena]], [[Stefano Giusto]], [[Rodolfo Russo]], [[Masaki Shigemori]], [[Nicholas P. Warner]], *Habemus Superstratum! A constructive proof of the existence of superstrata*, JHEP 1505 (2015) 110 [[arXiv:1503.01463](https://arxiv.org/abs/1503.01463)]

* [[Masaki Shigemori]], *Superstrata*, Gen Relativ Gravit **52** 51 (2020) [[arXiv:2002.01592](https://arxiv.org/abs/2002.01592), [doi:10.1007/s10714-020-02698-8](https://doi.org/10.1007/s10714-020-02698-8)]

category: people


Masako Bando

## Selected writings

On [[hidden local symmetry]] in [[chiral perturbation theory]]/[[quantum hadrodynamics]]:

* [[Masako Bando]], [[Taichiro Kugo]], Koichi Yamawaki, _Nonlinear realization and hidden local symmetries_, Physics Reports Volume 164, Issues 4–5, July 1988, Pages 217-314 (doi:10.1016/0370-1573(88)90019-1)

* [[Masako Bando]], [[Taichiro Kugo]], S. Uehara, Koichi Yamawaki, and T. Yanagida, _Is the $\rho$ Meson a Dynamical Gauge Boson of Hidden Local Symmetry?_, Phys. Rev. Lett. 54, 1215 (1985) ([doi:10.1103/PhysRevLett.54.1215](https://doi.org/10.1103/PhysRevLett.54.1215))

* [[Masako Bando]], Takanori Fujiwara, Koichi Yamawaki, _Generalized Hidden Local Symmetry and the $A_1$ Meson_, Progress of Theoretical Physics, Volume 79, Issue 5, May 1988, Pages 1140–1166 ([doi:10.1143/PTP.79.1140](https://doi.org/10.1143/PTP.79.1140))

category: people


Masako Takahashi

[[!redirects Masako Takashi]]

Masako Takahashi was a computer scientist at the Tokyo Institute of Technology.

* [Mathematical Genealogy page](https://www.mathgenealogy.org/id.php?id=180402)

## Selected writings

On the [[Church-Rosser theorem]]:

* [[Masako Takahashi]], *Parallel Reductions in $\lambda$-Calculus*, Journal of Symbolic Computation **7** 2  (1989) 113-123 [[doi:10.1006/inco.1995.1057](https://doi.org/10.1006/inco.1995.1057)]

category: people


Masamichi Takesaki

* [webpage](http://www.math.ucla.edu/~mt/)

category: people


Masanori Hanada

* [webpage](https://sites.google.com/site/hanadamasanori/home)

## Selected writings

On [[lattice gauge theory]]:

* [[Masanori Hanada]], _Markov Chain Monte Carlo for Dummies_ ([arXiv:1808.08490](https://arxiv.org/abs/1808.08490))

and in relation to [[string theory]]:

* {#Hanada16} [[Masanori Hanada]], _What lattice theorists can do for superstring/M-theory_,  International Journal of Modern Physics A Vol. 31, No. 22, 1643006 (2016)  ([arXiv:1604.05421](https://arxiv.org/abs/1604.05421))

On [[lattice gauge theory]] of [[super Yang-Mills theory]] via the [[BFSS matrix model]]:

* [[Masanori Hanada]], Paul Romatschke, _Lattice Simulations of 10d Yang-Mills toroidally compactified to 1d, 2d and 4d_ ([arXiv:1612.06395](https://arxiv.org/abs/1612.06395))

and via the  [[BMN matrix model]]:

* Hrant Gharibyan, [[Masanori Hanada]], Masazumi Honda, Junyu Liu, _Toward simulating Superstring/M-theory on a quantum computer_ ([arXiv:2011.06573](https://arxiv.org/abs/2011.06573))


Masanori Morishita

* [webpage](http://hyoka.ofc.kyushu-u.ac.jp/search/details/K002688/english.html)

## Selected writings

* {#KohnoMorishita06} [[Toshitake Kohno]], [[Masanori Morishita]] (eds.), _Primes and Knots_, Contemporary Mathematics, AMS 2006 ([conm:416](http://www.ams.org/bookstore-getitem/item=CONM-416))

* {#Morishita12} [[Masanori Morishita]], _Knots and Primes: An Introduction to Arithmetic Topology_, 2012 ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false))

## Related $n$Lab entries

* [[arithmetic topology]]

category: people


Masao Kitano

* [arXiv page](https://arxiv.org/a/kitano_m_1.html)

## Selected writings

On formulating [[classical field theory|classical]] [[electromagnetism]] and especially [[Maxwell's equations]] in terms of [[differential forms]] and the [[Hodge star operator]]:

* [[Masao Kitano]], *Reformulation of Electromagnetism with Differential Forms*, Chapter 2 in: *Trends in Electromagnetism -- From fundamentals to applications*, InTech (2012) 21-44 [[ISBN:978-953-51-0267-0](https://www.intechopen.com/books/1647), [[Kitano-ElectromagnetismWithForms.pdf:file]]]

category: people


Masatake Kuranishi

* [Wikipedia entry](http://de.wikipedia.org/wiki/Masatake_Kuranishi)

## Related entries

* [[Kuranishi deformation theory]]

category: people


Masatoshi Sato

* [institute page](https://ss.scphys.kyoto-u.ac.jp/TMS/en/members/profile_d01_sato/index.html)

* [GoogleScholar page](https://scholar.google.co.jp/citations?user=fl30hAgAAAAJ&hl=en)

## Selected writings

On the [[K-theory classification of topological phases of matter]] for [[topological crystalline insulators]]:

* [[Ken Shiozaki]], [[Masatoshi Sato]], [[Kiyonori Gomi]], *Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups*, Phys. Rev. B **95** (2017) 235425 $[$[arXiv:1701.08725](https://arxiv.org/abs/1701.08725), [doi:10.1103/PhysRevB.95.235425](https://doi.org/10.1103/PhysRevB.95.235425)$]$

and using the [[Atiyah-Hirzebruch spectral sequence]]:

* [[Ken Shiozaki]], [[Masatoshi Sato]], [[Kiyonori Gomi]], *Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups*, Phys. Rev. B **106** (2022) 165103 [[arXiv:1802.06694](https://arxiv.org/abs/1802.06694), [doi:10.1103/PhysRevB.106.165103](https://doi.org/10.1103/PhysRevB.106.165103)]

category: people


Maschke's theorem

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## Statement

\begin{theorem}\label{MaschkeTheorem}
**(Maschke's theorem)**
Given a [[finite group]] $G$ and a [[field]] $k$, the [[group algebra]] $k[G]$ is [[semisimple algebra|semisimple]] iff the [[characteristic]] of $k$ does not divide [[order of a group|order of the group]].
\end{theorem}

If the base $k$ is just a [[commutative unital ring]], then there is the following statement:

\begin{theorem}
\label{MaschkeOverRings}
If $|G|1_k$ is invertible in $k$, then an [[exact sequence]] of $k[G]$-[[modules]] [[split exact sequence|splits]] iff it splits after applying the [[forgetful functor]] from $k[G]$-modules to $k$-modules (and the splitting in ${}_{k[G]}Mod$ can be [[functor|functorially]] constructed from the splitting in ${}_{k}Mod$).
\end{theorem}

If $k$ is a [[field]], it follows that $k[G]$ is [[semisimple algebra|semisimple]], so that Thm. \ref{MaschkeOverRings} can be understood as a generalization of Maschke's theorem \ref{MaschkeTheorem}. This is also one of the motivations for the concept of *[[separable functors]]*.

The importance of the classical Maschke's theorem is that much is known about the structure of [[semisimple rings]] (starting with, e.g., [[Wedderburn's theorem]]).

## Related entries

* [[separable functor]]

## Literature

See also:

* Wikipedia, *[Maschke's theorem](http://en.wikipedia.org/wiki/Maschke%27s_theorem)*

[[!redirects Maschke theorem]]
[[!redirects Maschke's theorem]]
[[!redirects Maschke\'s theorem]]
[[!redirects Maschke's theorem]]


Maslov dequantization

Related entries include [[idempotent semiring]], [[quantization]], [[tropical geometry]], [[amoeba]]

* G. L. Litvinov, _The Maslov dequantization, idempotent and tropical mathematics: A brief introduction_, [math.GM/0507014](http://arxiv.org/abs/math/0507014)

> An idempotent version of real algebraic geometry was discovered in the report of O. Viro for the Barcelona Congress [172]. Starting from the idempotent correspondence principle O. Viro constructed a piecewise-linear geometry of polyhedra of a special kind in finite dimensional Euclidean spaces as a result of the Maslov dequantization of real algebraic geometry. He indicated important applications in real algebraic geometry (e.g., in the framework of Hilbert's 16th problem for constructing real algebraic varieties with prescribed properties and parameters) and relations to complex algebraic geometry and amoebas in the sense of I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky (see their book [61] and [173]). Then complex algebraic geometry was dequantized by G. Mikhalkin and the result turned out to be the same; this new 'idempotent' (or asymptotic) geometry is now often called the tropical algebraic geometry (...)

* related [nCafe comments](http://golem.ph.utexas.edu/category/2006/12/talks_at_higher_categories_and.html#c006892)
* Tropical and Idempotent Mathematics: International Workshop TROPICAL-07 ...By G. Grigorii Lazarevich Litvinov, S. N. Sergeev [gBooks](http://books.google.hr/books?id=o8oMzGywW1UC&lpg=PR9&ots=sXFQMy0W5Q&dq=maslov%20litvinov&pg=PR7#v=onepage&q=maslov%20litvinov&f=false)
* G. L. Litvinov, V.P. Maslov, _The correspondence principle for Idempotent Calculus and some computer applications_, In book Idempotency J. Gunawardena (Editor), Cambridge University Press, Cambridge, 1998, p.420-443

> ``There exists a (heuristic) correspondence, in the spirit of the correspondence principle in Quantum Mechanics, between important, useful and interesting constructions and results over the field of real (or complex) numbers (or the semiring of all nonnegative numbers) and similar constructions and results over idempotent semirings.''

* Oleg Viro's wiki: [Dequantization of real numbers](http://www.pdmi.ras.ru/~olegviro/dequant/2Quantization.html#SECTION00021000000000000000)


Maslov index

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## Overview

Consider a [[symplectic manifold]] (representing say a [[phase space]] of a [[physical system]]) of [[dimension]] $2n$  .

Recall that a [[Lagrangian submanifold]] is a smooth [[submanifold]] of dimension $n$ whose [[tangent spaces]] at all points are [[Lagrangian subspaces]], i.e. maximal [[isotropic subspaces]] with respect to the [[symplectic form]]. Lagrangian submanifold describes the phase of short-wave oscillations.

The _Maslov index_ is an invariant of a smooth path in a [[Lagrangian submanifold]].

The Maslov index can be reinterpreted as a [[characteristic class]] of theories of [[Lagrangian cobordism|Lagrangian and Legendrean cobordism]].

## Definition

### As a universal characteristic class

The first [[ordinary cohomology]] of the stable [[Lagrangian Grassmannian]] with [[integer]] [[coefficients]] is isomorphic to the [[integers]]

$$
  H^1(LGrass, \mathbb{Z})
  \simeq
  \mathbb{Z}
  \,.
$$

[[generalized the|The]] generator of this [[cohomology group]] is called the _universal Maslov index_

$$
  u \in H^1(LGrass, \mathbb{Z})
  \,.
$$

Since $LGrass$ is a [[classifying space]] for [[tangent bundles]] of [[Lagrangian submanifolds]], this is a [[universal characteristic class]] for Lagrangian submanifolds.

Specifically, given a [[Lagrangian submanifold]] $Y \hookrightarrow X$ of a [[symplectic manifold]] $(X,\omega)$, its [[tangent bundle]] is [[classifying space|classified]] by a function

$$
  i \;\colon\; Y \to LGrass
  \,.
$$

The _Maslov index of $Y$ is the universal Maslov index pulled back along this map

$$
  i^\ast u \in H^1(Y,\mathbb{Z})
  \,.
$$

## References and links

The index first appears maybe in

* [[Victor Maslov]], _Théorie des perturbations et méthodes asymptotiques_. 1972

Its cohomological interpretation as a [[universal characteristic class]] was explained in

* [[Vladimir Arnold]], _Characteristic class entering in quantization conditions_, Funct. Anal. its Appl. 1967, 1:1, 1–13, [doi](http://dx.doi.org/10.1007/BF01075861) (В. И. Арнольд, "О характеристическом классе, входящем в условия квантования", Функц. анализ и его прил., 1:1 (1967), 1–14, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2802&what=fullt&option_lang=rus))

A review in the context of [[geometric quantization]] (Maslov correction) is in

* Sean Bates, [[Alan Weinstein]], section 4.2 of _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)

The interpretation of the Maslov index as a quadratic space is due to

* T. Thomas. _The Maslov index as a quadratic space_. Math. Res. Lett. 13 no. 6 (2006), 985–999

and this definition and basic examples are briefly collected in

* [[Andrew Ranicki]], _The Maslov Index_, seminar notes 2010/11 ([pdf](http://www.maths.ed.ac.uk/~aar/maslovnotes.pdf))

See also

* G. Lion, [[Michele Vergne]], _The Weil representation, Maslov index and theta series_, Progress in Math. __6__, Birkhäuser 1980 (Rus. transl. Mir 1983).

* [[Alan Weinstein]], _The Maslov gerbe_, Lett. Math. Phys. __69__, 1-3, July, 2004, [doi](http://dx.doi.org/10.1007/s11005-004-0342-2). ([arXiv:0312274](http://arxiv.org/abs/math/0312274))

* [[Jean Leray]], _Lagrangian analysis and quantum mechanics. A mathematical structure related to asymptotic expansions and the Maslov index_, (trans. from French), MIT Press 1981. xvii+271 pp.

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Geometric asymptotics_, AMS 1977, [online](http://www.ams.org/online_bks/surv14);   _Semi-classical analysis_, 499 pages, [pdf](http://www-math.mit.edu/~vwg/semistart.pdf)

* J. J. Duistermaat, _On the Morse index in variational calculus_, Adv. Math. __21__ (1976), 2, 173--195, [pdf](http://www.maths.ed.ac.uk/~aar/papers/duistermaat.pdf).

* D. Salamon, E. Zehnder, _Morse theory for periodic solutions of Hamiltonian systems and the Maslov index_, Comm. Pure Appl. Math. __45__ (1992), no. 10, 1303--1360, [doi](http://dx.doi.org/10.1002/cpa.3160451004)

* [[Joel Robbin]], [[Dietmar Salamon]], _The Maslov index for paths_, Topology __32__ (1993), no. 4, 827--844, ([doi](http://dx.doi.org/10.1016/0040-9383%2893%2990052-W); preprint version [pdf](http://www.math.ethz.ch/~salamon/PREPRINTS/maslov.pdf)); _The spectral flow and the Maslov index_, Bull. London Math. Soc. __27__ (1995), no. 1, 1--33 ([doi](http://dx.doi.org/10.1112/blms/27.1.1))

* A. B. Givental', _Global properties of the Maslov index and Morse theory_, Funct. Anal. Its. Appl. __22__, 2, 1988, [doi](http://dx.doi.org/10.1007/BF01077609) (Rus. orig: функц. анализ и его приложения __22__, 1988, вып. 2, 69—70: [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1113&volume=22&year=1988&issue=2&fpage=69&what=fullt&option_lang=eng))

* A. B. Giventalʹ, _The nonlinear Maslov index_, in "Geometry of low-dimensional manifolds" vol. 2 (Durham, 1989), 35--43,
London Math. Soc. Lec. Note Ser. __151__, Cambridge Univ. Press 1990.

* Maurice A. de Gosson, _Maslov classes, metaplectic representation and Lagrangian quantization_, Math. Research __95__, Akademie-Verlag, Berlin, 1997. 186 pp.; _Symplectic geometry, Wigner-Weyl-Moyal calculus, and quantum mechanics in phase space_, 385 pp. [pdf](http://opus.kobv.de/ubp/volltexte/2009/3021/pdf/2006_06.pdf)

* Leo T. Butler, _The Maslov cocycle, smooth structures and real-analytic complete integrability_, [arxiv/0708.3157](http://uk.arxiv.org/abs/0708.3157v2)

* S. Merigon, _L'indice de Maslov en dimension infinie_, [J. Lie Theory 18](http://www.heldermann.de/JLT/JLT18/JLT181/jlt18010.htm) (2008), no. 1, 161--180.

* S. E. Cappell, R. Lee, E. Y. Miller, _On the Maslov index_, Comm. Pure Appl. Math. __47__ (1994), no. 2, 121--186.

* K. Furutani, _Fredholm–Lagrangian–Grassmannian and the Maslov index_, J. Geom. Phys. __51__, 3, July 2004, 269--331, [doi](http://dx.doi.org/10.1016/j.geomphys.2004.04.001)

* Many links are at [[Andrew Ranicki]]'s [Maslov index seminar](http://www.maths.ed.ac.uk/~aar/maslov.htm) page.

* Paolo Piccione, Daniel Victor Tausk, _A student's guide to symplectic spaces, Grassmannians and Maslov index_, IMPA 2011, [pdf](http://www.impa.br/opencms/pt/biblioteca/pm/PM_27.pdf)

* M. V. Finkelberg, _Orthogonal Maslov index_, Funct. Anal. Appl. 29(1) 72–74 (1995) [doi](https://doi.org/10.1007/bf01077047)
* [[Alan Weinstein]], _The Maslov cycle as a Legendre singularity and projection of a wavefront set_, Bull. Braz. Math. Soc., N.S. 44, 593--610 (2013) [doi](https://doi.org/10.1007/s00574-013-0026-6)

Application in the theory of Schroedinger operators:

* Yuri Latushkin, Alim Sukhtayev, Selim Sukhtaiev, _The Morse and Maslov indices for Schrödinger operators_, [arxiv/1411.1656](http://arxiv.org/abs/1411.1656); Yuri Latushkin, Alim Sukhtayev, _Hadamard-type formulas via the Maslov form_, [arxiv/1601.07509](http://arxiv.org/abs/1601.07509)
[[!redirects Maslov class]]


Maslov-Feynman calculus

__Maslov-Feynman calculus__ (Feynman-Maslov commutator expansion) is an expansion in terms of commutators of expressions in associative algebras. It has been pioneered independently by [[Richard Feynman]] in quantum field theory and by [[V. P. Maslov]] in the analysis of operators (and, in particular, WKB method).

Kapranov used it in "[[Kapranov's noncommutative geometry]] based on commutator expansions".

[[!redirects Feynman-Maslov commutator expansion]]
[[!redirects Feynman-Maslov calculus]]


Masoud Khalkhali

__Masoud Khalkhali__ is Iranian-Canadian mathematician with professorship at the University of Western Ontario in London, Ontario. His main focus is on [[noncommutative geometry]] and homological algebra, especially the [[cyclic homology]]. He is one of the contrubutors to the group blog <http://noncommutativegeometry.blogspot.com>.

* [web](http://www.math.uwo.ca/~masoud/cv)

* _A short survey of cyclic cohomology_, [arxiv/1008.1212](http://arxiv.org/abs/1008.1212)

* book: _Basic Noncommutative Geometry_, Europ. Math. Soc. 2009, 239 p. [ad](http://www.ems-ph.org/books/book.php?proj_nr=109)

[[!redirects M. Khalkhali]]


mass

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# Mass
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{: toc}

## Definitions

The __mass__ of a [[physics|physical]] system is its intrinsic [[energy]].

It is a somewhat arbitrary choice, made necessary by the theory of [[relativity]], to decide whether the old term 'mass' should be used for intrinsic energy or total energy; one may clarify that intrinsic energy is __rest mass__ while total energy is __relativistic mass__.

Traditionally, mass and energy are measured in different units related by the [[speed of light]]: $E = m c^2$.  Thus one may call a system\'s mass its __rest energy__ when it is measured in units of energy; conversely, one may call a systems\'s total energy its __relativistic mass__ when it is measured in units of mass.

The __[[kinetic energy]]__ of a system is the difference between its total energy and its mass (or rest energy).  Thus the kinetic energy is the energy due to the motion of the system (or better, due to the relative motion of the system and the observer).

When a system is thought of as the combination of several [[subsystem|subsystems]], then its mass also may be broken into several pieces:

*  for each subsystem, its mass,
*  for each subsystem, its kinetic energy due to relative motion of the subsystem,
*  for each pair of subsystems, the [[potential energy]] of their interaction.

The latter entry may be positive (in which case the system is liable to separate into its component parts, at least at sufficiently high [[temperature]], giving off energy) or negative (in which case the system is __bound__ and can be expected to remain together until additional energy is added).

## Formulas

Mass $m$ may be given in terms of energy $E$ and [[linear momentum]] $\mathbf{p}$ as
$$ m = \sqrt {E^2 - {|\mathbf{p}|}^2} .$$
As $\mathbf{p} = E \mathbf{v}$ for $\mathbf{v}$ the linear [[velocity]], we also have
$$ m = E \sqrt {1 - {|\mathbf{v}|}^2} .$$
(For nonrelativistic units, change $E$ to $E/c^2$, $\mathbf{p}$ to $\mathbf{p}/c$, and $\mathbf{v}$ to $\mathbf{v}/c$.)

For a particle travelling at the [[speed of light]], we therefore have $m = 0$.  When $\mathbf{v} = 0$, we have $m = E$.  When $0 \lt {|\mathbf{v}|} \ll c$, we still have $ m \approx E$.  However, in nonrelativistic physics, it is much more useful to use [[kinetic energy]] $K = E - m$ instead of the total energy $E$; then we have
$$ m = K \frac { 1 - {|\mathbf{v}|}^2 + \sqrt {1 - {|\mathbf{v}|^2}} } { {|\mathbf{v}|}^2 } $$
exactly and
$$ m \approx 2 \frac K { {|\mathbf{v}|}^2 } $$
when $0 \lt {|\mathbf{v}|} \ll c$.  Notice that this last expression makes sense already in nonrelativistic units.

For a system made up of subsystems (indexed by $i,j,\ldots$), we have
$$ m = \sum_i m_i + \sum_i \tilde{K}_i + \sum_{i,j} U_{i,j} ,$$
where $m_i$ is the mass of subsystem $i$, $\tilde{K}_i$ is the kinetic energy of subsystem $i$ relative to the whole system, and $U_{i,j}$ is the potential energy of the [[forces]] between system $i$ and system $j$.  The sum
$$ - \sum_{i,j} U_{i,j} = \sum_i m_i + \sum_i \tilde{K}_i - m $$
is the __[[binding energy]]__ of the system; when it is positive, it gives the minimum energy that must be added to the system to break it into components that do not interact.  In many cases, the relative kinetic energies $\tilde{K}_i$ are negligible, so we can calculate binding energy by subtracting masses as
$$ - \sum_{i,j} U_{i,j} \approx \sum_i m_i - m .$$
In nonrelativistic physics, where kinetic energy is always negligible, the $U_{i,j}$ are *also* negligible compared to the masses, so it is not possible at all to calculate the sign of the binding energy using masses.  Instead, we have
$$ m \approx \sum_i m_i ,$$
the statement of nonrelativistic __conservation of mass__.

Many of the formulas above rely on ${|\mathbf{v}|} \leq c$, or equivalently ${|\mathbf{p}|} \leq E$.  For [[tachyon|tachyons]], where this is violated, the mass becomes an [[imaginary number]].  For this reason, it may make more sense to work with $m^2$ (which is always real) than with $m$ itself.  We also implicitly assume that $E \gt 0$; for [[exotic particle|exotic particles]] in which $E \lt 0$, it is convenient also to take $m \lt 0$.  (In this case, of course, it is *not* sufficient to work only with $m^2$.)

## Related entries

* [[mass term]]

* [[mass gap]]

* [[Higgs mechanism]]

* [[ADM mass]]

* [[massive Yang-Mills theory]]

* [[space]], [[time]]

* [[spacetime]]

[[!include fundamental scales -- contents]]

## References
 {#References}

See also:

* Wikipedia, *[Mass](https://en.wikipedia.org/wiki/Mass)*

[[!redirects mass]]
[[!redirects masses]]

[[!redirects rest mass]]
[[!redirects rest masses]]