integers type

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
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[[!include type theory - contents]]
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#### Deduction and Induction
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[[!include deduction and induction - contents]]
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#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
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# Contents
* table of contents
{: toc}

## Idea

In [[type theory]]: the _integers type_ is the [[type]] of [[integers]].

## Definitions

### As the inductive type generated by an element and an equivalence of types

Assuming that [[identification types]], [[equivalence types]] and [[dependent product types]] exist in the type theory, the integers type is the [[inductive type]] generated by an element and an [[equivalence of types]]:

Formation rules for the integers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{Z} \; \mathrm{type}}$$

Introduction rules for the integers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{Z}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash s:\mathbb{Z} \simeq \mathbb{Z}}$$

Elimination rules for the integers type:
$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{Z}} C(x) \simeq C(s(x)) \quad \Gamma \vdash n:\mathbb{Z}}{\Gamma \vdash \mathrm{ind}_\mathbb{Z}^C(c_0, c_s, n):C(n)}$$

Computation rules for the integers type:

* Judgmental computation rules

$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{Z}} C(x) \simeq C(s(x))}{\Gamma \vdash \mathrm{ind}_\mathbb{Z}^C(c_0, c_s, 0) \equiv c_0:C(0)}$$

$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{Z}} C(x) \simeq C(s(x)) \quad \Gamma \vdash n:\mathbb{Z}}{\Gamma \vdash \mathrm{ind}_\mathbb{Z}^C(c_0, c_s, s(n)) \equiv c_s(n)(\mathrm{ind}_\mathbb{Z}^C(c_0, c_s, n)):C(s(n))}$$

* Typal computation rules

$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{Z}} C(x) \simeq C(s(x))}{\Gamma \vdash \beta_\mathbb{Z}^0(c_0, c_s):\mathrm{Id}_{C(0)}(\mathrm{ind}_\mathbb{Z}^C(c_0, c_s, 0), c_0)}$$

$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{Z}} C(x) \simeq C(s(x)) \quad \Gamma \vdash n:\mathbb{Z}}{\Gamma \vdash \beta_\mathbb{Z}^s(c_0, c_s, n):\mathrm{Id}_{C(s(n))}(\mathrm{ind}_\mathbb{Z}^C(c_0, c_s, s(n)), c_s(n)(\mathrm{ind}_\mathbb{Z}^C(c_0, c_s, n)))}$$

Uniqueness rules for the integers type:

* Judgmental uniqueness rules

$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathbb{Z}} C(x) \quad \Gamma \vdash n:\mathbb{Z}}{\Gamma \vdash \mathrm{ind}_\mathbb{Z}^C(c(0), \lambda x:\mathbb{Z}.c(s(x)), n) \equiv c(n):C(n)}$$

* Typal uniqueness rules

$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathbb{Z}} C(x) \quad \Gamma \vdash n:\mathbb{Z}}{\Gamma \vdash \eta_\mathbb{Z}(c, n):\mathrm{Id}_{C(n)}(\mathrm{ind}_\mathbb{Z}^C(c(0), \lambda x:\mathbb{Z}.c(s(x)), n), c(n))}$$

The elimination, computation, and uniqueness rules for the integers type state that the integers type satisfy the **dependent universal property of the integers**. If the dependent type theory also has [[dependent sum types]] and [[product types]], allowing one to define the [[uniqueness quantifier]], the dependent universal property of the integers could be simplified to the following rule:

$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{Z}} C(x) \simeq C(s(x))}{\Gamma \vdash \mathrm{up}_\mathbb{Z}^C(c_0, c_s):\exists!c:\prod_{x:\mathbb{Z}} C(x).\mathrm{Id}_{C(0)}(c(0), c_0) \times \prod_{x:\mathbb{Z}} \mathrm{Id}_{C(s(x))}(c(s(x)), c_s(c(x)))}$$

### As the underlying type of the free infinity-group on the unit type

The integers type is the [[underlying type of free infinity-group]] on the [[unit type]]

$$\mathbb{Z} \coloneqq \mathrm{UnderlyingTypeOfFree}\infty\mathrm{Group}(\mathbb{1})$$

### As the loop space of the circle type

Given a [[dependent type theory]] with [[identity types]], [[equivalence types]], a [[univalent universe]], and the [[circle type]] which is an [[essentially small type]] relative to the universe universe, the type of integers is defined as the [[loop space type]] of the circle type at the canonical element $\mathrm{base}:S^1$:

$$\mathbb{Z} \coloneqq \mathrm{base} =_{S^1} \mathrm{base}$$

### As a quotient of a product type

+-- {: .num_defn}
###### Definition
The integers type is defined as the [[higher inductive type]] generated by:

 * A function $inj : \mathbf{2} \times \mathbb{N} \rightarrow \mathbb{Z}$.
 * An identity representing that positive and negative zero are equal: $\nu_0: inj(0, 0) = inj(1, 0)$.
=--

+-- {: .num_defn}
###### Definition
The integers type is defined as the [[higher inductive type]] generated by:

 * A function $inj : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{Z}$.
 * A dependent product of functions between identities representing that equivalent differences are equal:
$$equivdiff : \prod_{a:\mathbb{N}} \prod_{b:\mathbb{N}} \prod_{c:\mathbb{N}} \prod_{d:\mathbb{N}} (a + d = c + b) \to (inj(a,b) = inj(c,d))$$
 * A set-truncator
$$\tau_0: \prod_{a:\mathbb{Z}} \prod_{b:\mathbb{Z}} isProp(a=b)$$
=--

## Properties

We assume in this section that the integers type is defined as the inductive type generated by an element and an [[equivalence of types]].

### Extensionality principle of the integers type

By the recursion principle of the natural numbers, there is a function

$$\mathrm{rec}_{\mathbb{N}}(0, \pi_1(s)):\mathbb{N} \to \mathbb{Z}$$

which takes zero of the natural numbers to zero of the integers and the successor function of the natural numbrers to the successor equivalence of the integers, where $\pi_1$ is the first projection function of the dependent sum type $\sum_{f:\mathbb{Z} \to \mathbb{Z}} \mathrm{isEquiv}(f)$. The extensionality principle of the circle type states that the above function is an [[embedding of types]],
$$\mathrm{ext}_{S^1}:\mathrm{isEmbed}\left(\mathrm{rec}_{\mathbb{N}}(0, \pi_1(s))\right)$$
or equivalently that application of the above function is an equivalence for all natural numbers $m:\mathbb{N}$, $n:\mathbb{N}$

$$\mathrm{ext}_{S^1}:\prod_{m:\mathbb{N}} \prod_{n:\mathbb{N}} \mathrm{isEquiv}\left(\mathrm{ap}_{\mathrm{rec}_{\mathbb{N}}(0, \pi_1(s))}(m, n)\right)$$

\begin{theorem}
The integers type has decidable equality.
\end{theorem}

\begin{proof}
...
\end{proof}

### Double induction

From the inference rules for the integers type, one could derive the following rules for double induction, which is important for the definition of addition, subtraction, and multiplication on the integers:

$$\frac{
\begin{array}{c}
\Gamma, x:\mathbb{Z}, y:\mathbb{Z} \vdash C(x, y) \; \mathrm{type} \quad \Gamma \vdash c_{0, 0}:C(0, 0) \quad \Gamma \vdash c_{0, s}:\prod_{x:\mathbb{Z}} C(0, x) \simeq C(0, s(x)) \\
\Gamma \vdash c_{s, 0}:\prod_{x:\mathbb{Z}} C(x, 0) \simeq C(s(x), 0) \quad \Gamma \vdash c_{s, s}:\prod_{x:\mathbb{Z}} \prod_{y:\mathbb{Z}} C(x, y) \simeq C(s(x), s(y))
\end{array}
}{\Gamma \vdash \mathrm{ind}_\mathbb{Z}^C(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}):\prod_{m:\mathbb{Z}} \prod_{n:\mathbb{Z}} C(m, n)}$$

$$\frac{
\begin{array}{c}
\Gamma, x:\mathbb{Z}, y:\mathbb{Z} \vdash C(x, y) \; \mathrm{type} \quad \Gamma \vdash c_{0, 0}:C(0, 0) \quad \Gamma \vdash c_{0, s}:\prod_{x:\mathbb{Z}} C(0, x) \simeq C(0, s(x)) \\
\Gamma \vdash c_{s, 0}:\prod_{x:\mathbb{Z}} C(x, 0) \simeq C(s(x), 0) \quad \Gamma \vdash c_{s, s}:\prod_{x:\mathbb{Z}} \prod_{y:\mathbb{Z}} C(x, y) \simeq C(s(x), s(y))
\end{array}
}{\Gamma \vdash \beta_\mathbb{Z}^{C, 0, 0}(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}):\mathrm{ind}_\mathbb{Z}^C(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}, 0, 0) =_{C(0, 0)} c_{0, 0}}$$

$$\frac{
\begin{array}{c}
\Gamma, x:\mathbb{Z}, y:\mathbb{Z} \vdash C(x, y) \; \mathrm{type} \quad \Gamma \vdash c_{0, 0}:C(0, 0) \quad \Gamma \vdash c_{0, s}:\prod_{x:\mathbb{Z}} C(0, x) \simeq C(0, s(x)) \\
\Gamma \vdash c_{s, 0}:\prod_{x:\mathbb{Z}} C(x, 0) \simeq C(s(x), 0) \quad \Gamma \vdash c_{s, s}:\prod_{x:\mathbb{Z}} \prod_{y:\mathbb{Z}} C(x, y) \simeq C(s(x), s(y))
\end{array}
}{\Gamma \vdash \beta_\mathbb{Z}^{C, 0, s}(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}):\prod_{n:\mathbb{Z}} \mathrm{ind}_\mathbb{Z}^C(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}, 0, s(n)) =_{C(0, s(n))} c_{0, s}(n, \mathrm{ind}_\mathbb{Z}^C(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}, 0, n))}$$

$$\frac{
\begin{array}{c}
\Gamma, x:\mathbb{Z}, y:\mathbb{Z} \vdash C(x, y) \; \mathrm{type} \quad \Gamma \vdash c_{0, 0}:C(0, 0) \quad \Gamma \vdash c_{0, s}:\prod_{x:\mathbb{Z}} C(0, x) \simeq C(0, s(x)) \\
\Gamma \vdash c_{s, 0}:\prod_{x:\mathbb{Z}} C(x, 0) \simeq C(s(x), 0) \quad \Gamma \vdash c_{s, s}:\prod_{x:\mathbb{Z}} \prod_{y:\mathbb{Z}} C(x, y) \simeq C(s(x), s(y))
\end{array}
}{\Gamma \vdash \beta_\mathbb{Z}^{C, s, 0}(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}):\prod_{n:\mathbb{Z}} \mathrm{ind}_\mathbb{Z}^C(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}, s(n), 0) =_{C(s(n), 0)} c_{s, 0}(n, \mathrm{ind}_\mathbb{Z}^C(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}, n, 0))}$$

$$\frac{
\begin{array}{c}
\Gamma, x:\mathbb{Z}, y:\mathbb{Z} \vdash C(x, y) \; \mathrm{type} \quad \Gamma \vdash c_{0, 0}:C(0, 0) \quad \Gamma \vdash c_{0, s}:\prod_{x:\mathbb{Z}} C(0, x) \simeq C(0, s(x)) \\
\Gamma \vdash c_{s, 0}:\prod_{x:\mathbb{Z}} C(x, 0) \simeq C(s(x), 0) \quad \Gamma \vdash c_{s, s}:\prod_{x:\mathbb{Z}} \prod_{y:\mathbb{Z}} C(x, y) \simeq C(s(x), s(y))
\end{array}
}{\Gamma \vdash \beta_\mathbb{Z}^{C, s, s}(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}):\prod_{m:\mathbb{Z}} \prod_{n:\mathbb{Z}} \mathrm{ind}_\mathbb{Z}^C(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}, s(m), s(n)) =_{C(s(m), s(n))} c_{0, s}(m, n, \mathrm{ind}_\mathbb{Z}^C(c_{0, 0}, c_{0, s}, c_{s, 0}, c_{s, s}, m, n))}$$

### Addition on the integers

\begin{definition}
The binary operation **addition** is defined by double induction, where the type family $C$ is the constant type family $\mathbb{Z}$, $c_{0, 0}$ is $0$, $c_{0, s}$ and $c_{s, 0}$ are both the constant function $\lambda n:\mathbb{Z}.s$ which takes integers to the successor equivalence, and $c_{s, s}$ is the constant function $\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.s \circ s$ which takes pairs of integers to the composition of the successor equivalence with itself, and is defined pointwise as

$$x:\mathbb{Z}, y:\mathbb{Z} \vdash x + y \equiv \mathrm{ind}_\mathbb{Z}^\mathbb{Z}(0, \lambda n:\mathbb{Z}.s, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.s \circ s, x, y):\mathbb{Z}$$

We have similar definitions for the computation rules of addition:

$$\beta_\mathbb{Z}^{+, 0, 0} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, 0, 0}(0, \lambda n:\mathbb{Z}.s, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.s \circ s):0 + 0 =_{\mathbb{Z}} 0$$

$$\beta_\mathbb{Z}^{+, 0, s} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, 0, s}(0, \lambda n:\mathbb{Z}.s, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.s \circ s):\prod_{x:\mathbb{Z}} 0 + s(x) =_{\mathbb{Z}} ((\lambda n:\mathbb{Z}.s)(x))(0 + x)$$

$$\beta_\mathbb{Z}^{+, s, 0} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, s, 0}(0, \lambda n:\mathbb{Z}.s, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.s \circ s):\prod_{x:\mathbb{Z}} s(x) + 0 =_{\mathbb{Z}} ((\lambda n:\mathbb{Z}.s)(x))(x + 0)$$

$$\beta_\mathbb{Z}^{+, s, s} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, s, s}(0, \lambda n:\mathbb{Z}.s, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.s \circ s):\prod_{x:\mathbb{Z}} \prod_{y:\mathbb{Z}} s(x) + s(y) =_{\mathbb{Z}} ((\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.s \circ s)(x, y))(x + y)$$
\end{definition}

\begin{theorem}
The integers type is a non-coherent [[H-space]] with respect to the zero integer $0:\mathbb{Z}$ found in the [[introduction rule]] of the integers type and addition $x:\mathbb{Z}, y:\mathbb{Z} \vdash x + y:\mathbb{Z}$ defined above.
\end{theorem}

\begin{proof}
We define the [[homotopies]] for the [[H-space]] structure of the integers type by induction on the integers type. For the case with the element $0:\mathbb{Z}$, one gets the identity $\beta_\mathbb{Z}^{+, 0, 0}:0 + 0 =_\mathbb{Z} 0$ from the rules for addition. For the case with the equivalence $s:\mathbb{Z} \simeq \mathbb{Z}$, the application to $s$ is itself a dependent function of a family of equivalences,
$$\mathrm{ap}_{s}:\prod_{y:\mathbb{Z}} \prod_{x:\mathbb{Z}} (y =_\mathbb{Z} x) \simeq (s(y) =_\mathbb{Z} s(x))$$

Substituting $0 + x$ and $x + 0$ in for $y$ yields the family of equivalences

$$x:\mathbb{Z} \vdash \mathrm{ap}_{s}(0 + x, x):(0 + x =_\mathbb{Z} x) \simeq (s(0 + x) =_\mathbb{Z} s(x))$$

$$x:\mathbb{Z} \vdash \mathrm{ap}_{s}(x + 0, x):(x + 0 =_\mathbb{Z} x) \simeq (s(x + 0) =_\mathbb{Z} s(x))$$

and by the [[introduction rule]] of [[dependent product types]], one gets the dependent functions of families of equivalences

$$\lambda x:\mathbb{Z}.\mathrm{ap}_{s}(0 + x, x):\prod_{x:\mathbb{Z}} (0 + x =_\mathbb{Z} x) \simeq (s(0 + x) =_\mathbb{Z} s(x))$$

$$\lambda x:\mathbb{Z}.\mathrm{ap}_{s}(x + 0, x):\prod_{x:\mathbb{Z}} (x + 0 =_\mathbb{Z} x) \simeq (s(x + 0) =_\mathbb{Z} s(x))$$

and by the elimination rules for the integers type, one has families of identifications

$$x:\mathbb{Z} \vdash \mathrm{ind}_\mathbb{Z}^{0 + x =_\mathbb{Z} x}(\beta_\mathbb{Z}^{+, 0, 0}, \lambda x:\mathbb{Z}.\mathrm{ap}_{s}(0 + x, x), x):0 + x =_\mathbb{Z} x$$
$$x:\mathbb{Z} \vdash \mathrm{ind}_\mathbb{Z}^{x + 0 =_\mathbb{Z} x}(\beta_\mathbb{Z}^{+, 0, 0}, \lambda x:\mathbb{Z}.\mathrm{ap}_{s}(x + 0, x), x):x + 0 =_\mathbb{Z} x$$

By the [[introduction rule]] of [[dependent product types]], one gets [[homotopies]]

$$\mathrm{lunitadd}_\mathbb{Z} \equiv \lambda x:\mathbb{Z}.\mathrm{ind}_\mathbb{Z}^{0 + x =_\mathbb{Z} x}(\beta_\mathbb{Z}^{+, 0, 0}, \lambda x:\mathbb{Z}.\mathrm{ap}_{s}(0 + x, x), x):\prod_{x:\mathbb{Z}} 0 + x =_\mathbb{Z} x$$

$$\mathrm{runitadd}_\mathbb{Z} \equiv \lambda x:\mathbb{Z}.\mathrm{ind}_\mathbb{Z}^{x + 0 =_\mathbb{Z} x}(\beta_\mathbb{Z}^{+, 0, 0}, \lambda x:\mathbb{Z}.\mathrm{ap}_{s}(x + 0, x), x):\prod_{x:\mathbb{Z}} x + 0 =_\mathbb{Z} x$$

Thus, the integers type is a non-coherent [[H-space]].
\end{proof}

\begin{theorem}
The integers type is an associative non-coherent H-space.
\end{theorem}

\begin{proof}
...
\end{proof}

\begin{theorem}
The integers type is invertible with respect to the H-space structure of addition - such that given an integer $x:\mathbb{Z}$, left addition and right addition by $x$ is each an [[equivalence of types]].
\end{theorem}

\begin{proof}
...
\end{proof}

\begin{corollary}
The integers type is a non-coherent [[H-group]], a non-coherent [[H-space]] which is invertible and associative.
\end{corollary}

\begin{theorem}
The integers type is non-coherently abelian with respect to addition.
\end{theorem}

\begin{proof}
...
\end{proof}

### Negation on the integers

\begin{definition}
The unary operation **negation** is defined by induction on the integers type, where the type family $C$ is the constant type family $\mathbb{Z}$, $c_{0}$ is $0$, and $c_{s}$ is the constant function $\lambda n:\mathbb{Z}.s^{-1}$ which takes integers to the inverse of the successor equivalence, and is defined pointwise as

$$x:\mathbb{Z} \vdash -x \equiv \mathrm{ind}_\mathbb{Z}^\mathbb{Z}(0, \lambda n:\mathbb{Z}.s^{-1}, x):\mathbb{Z}$$

We have similar definitions for the computation rules of negation:

$$\beta_\mathbb{Z}^{-, 0} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, 0}(0, \lambda n:\mathbb{Z}.s^{-1}):-0 =_{\mathbb{Z}} 0$$

$$\beta_\mathbb{Z}^{-, s} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, s}(0, \lambda n:\mathbb{Z}.s^{-1}):\prod_{x:\mathbb{Z}} -s(x) =_{\mathbb{Z}} ((\lambda n:\mathbb{Z}.s^{-1})(x))(-x) \equiv s^{-1}(-x)$$
\end{definition}

### Subtraction on the integers

\begin{definition}
The binary operation **subtraction** is defined by double induction, where the type family $C$ is the constant type family $\mathbb{Z}$, $c_{0, 0}$ is $0$, $c_{0, s}$ is the constant function $\lambda n:\mathbb{Z}.s^{-1}$ which takes integers to the inverse of the successor equivalence, $c_{s, 0}$ is the constant function $\lambda n:\mathbb{Z}.s$ which takes integers to the successor equivalence, and $c_{s, s}$ is the constant function $\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}$ which takes pairs of integers to the [[identity equivalence]] on the integers, and is defined pointwise as

$$x:\mathbb{Z}, y:\mathbb{Z} \vdash x - y \equiv \mathrm{ind}_\mathbb{Z}^\mathbb{Z}(0, \lambda n:\mathbb{Z}.s^{-1}, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, x, y):\mathbb{Z}$$

We have similar definitions for the computation rules of subtraction:

$$\beta_\mathbb{Z}^{-, 0, 0} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, 0, 0}(0, \lambda n:\mathbb{Z}.s^{-1}, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}):0 - 0 =_{\mathbb{Z}} 0$$

$$\beta_\mathbb{Z}^{-, 0, s} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, 0, s}(0, \lambda n:\mathbb{Z}.s^{-1}, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}):\prod_{x:\mathbb{Z}} 0 - s(x) =_{\mathbb{Z}} ((\lambda n:\mathbb{Z}.s^{-1})(x))(0 - x) \equiv s^{-1}(0 - x)$$

$$\beta_\mathbb{Z}^{-, s, 0} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, s, 0}(0, \lambda n:\mathbb{Z}.s^{-1}, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}):\prod_{x:\mathbb{Z}} s(x) - 0 =_{\mathbb{Z}} ((\lambda n:\mathbb{Z}.s)(x))(x - 0) \equiv s(x - 0)$$

$$\beta_\mathbb{Z}^{-, s, s} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, s, s}(0, \lambda n:\mathbb{Z}.s^{-1}, \lambda n:\mathbb{Z}.s, \lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}):\prod_{x:\mathbb{Z}} \prod_{y:\mathbb{Z}} s(x) - s(y) =_{\mathbb{Z}} ((\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z})(x, y))(x - y) \equiv x - y$$
\end{definition}

\begin{theorem}
For all $x:\mathbb{Z}$ and $y:\mathbb{Z}$, we have an identification of type
$$x - y =_\mathbb{Z} x + (- y)$$
\end{theorem}

### Set-truncation structure

To do: figure out a way to prove that the integers type is [[0-truncated]] without using the [[natural numbers type]]. Then we define the natural numbers type $\mathbb{N}$ as the quotient of the integers type by its [[group of units]] $\mathbb{Z}/\mathbb{Z}^\times$.

\begin{theorem}
The [[natural numbers type]] $\mathbb{N}$ is [[0-truncated]].
\end{theorem}

\begin{proof}
...
\end{proof}

\begin{theorem}
The natural numbers type $\mathbb{N}$ and the integers type $\mathbb{Z}$ are equivalent to each other.
\end{theorem}

\begin{proof}
...
\end{proof}

\begin{corollary}
The integers type are an abelian group.
\end{corollary}

\begin{proof}
Since $\mathbb{N}$ is [[0-truncated]], and equivalences preserve truncatedness, by the equivalence $e:\mathbb{N} \simeq \mathbb{Z}$, $\mathbb{Z}$ is also 0-truncated. Since $\mathbb{Z}$ is an non-coherently abelian [[H-group]], this implies that $\mathbb{Z}$ is a [[abelian group]].
\end{proof}

### Multiplication on the integers

To do: figure out a way to prove the various properties of multiplication without using the fact that the integers are a set (i.e. the integers are a non-coherently abelian H-ring).

\begin{definition}
Since the integers are a non-coherently abelian H-group with respect to addition, right addition by an integer $k$, $\lambda z:\mathbb{Z}.z + k$, is an equivalence.

The binary operation **multiplication** is defined by double induction, where the type family $C$ is the constant type family $\mathbb{Z}$, $c_{0, 0}$ is $0$, $c_{0, s}$ and $c_{s, 0}$ are both the constant function $\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}$ which takes integers to the [[identity equivalence]] on the integers, and $c_{s, s}$ is the function $\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\lambda z:\mathbb{Z}.z + s(m + n)$ which takes pairs of integers to the equivalence $\lambda z:\mathbb{Z}.z + s(m + n)$, and is defined pointwise as

$$x:\mathbb{Z}, y:\mathbb{Z} \vdash x \cdot y \equiv \mathrm{ind}_\mathbb{Z}^\mathbb{Z}(0, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda m:\mathbb{Z}.\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\lambda z:\mathbb{Z}.z + s(m + n), x, y):\mathbb{Z}$$

We have similar definitions for the computation rules of multiplication:

$$\beta_\mathbb{Z}^{\cdot, 0, 0} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, 0, 0}(0, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda m:\mathbb{Z}.\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\lambda z:\mathbb{Z}.z + s(m + n)):0 \cdot 0 =_{\mathbb{Z}} 0$$

$$\beta_\mathbb{Z}^{\cdot, 0, s} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, 0, s}(0, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda m:\mathbb{Z}.\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\lambda z:\mathbb{Z}.z + s(m + n)):\prod_{x:\mathbb{Z}} 0 \cdot s(x) =_{\mathbb{Z}} ((\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z})(x))(0 \cdot x) \equiv 0 \cdot x$$

$$\beta_\mathbb{Z}^{\cdot, s, 0} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, s, 0}(0, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda m:\mathbb{Z}.\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\lambda z:\mathbb{Z}.z + s(m + n)):\prod_{x:\mathbb{Z}} s(x) \cdot 0 =_{\mathbb{Z}} ((\lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z})(x))(x \cdot 0) \equiv x \cdot 0$$

$$\beta_\mathbb{Z}^{\cdot, s, s} \equiv \beta_\mathbb{Z}^{\mathbb{Z}, s, s}(0, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda n:\mathbb{Z}.\mathrm{id}_\mathbb{Z}, \lambda m:\mathbb{Z}.\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\lambda z:\mathbb{Z}.z + s(m + n)):\prod_{x:\mathbb{Z}} \prod_{y:\mathbb{Z}} s(x) \cdot s(y) =_{\mathbb{Z}} ((\lambda m:\mathbb{Z}.\lambda n:\mathbb{Z}.\lambda z:\mathbb{Z}.z + s(m + n)))(x, y))(x \cdot y) \equiv x \cdot y + s(x + y)$$
\end{definition}

\begin{theorem}
The integers type is an [[absorption magma]] with respect to the zero integer $0:\mathbb{Z}$ and multiplication $x:\mathbb{Z}, y:\mathbb{Z} \vdash x \cdot y:\mathbb{Z}$ defined above.
\end{theorem}

\begin{proof}
We begin with the left absorption law, that $0 \cdot x =_\mathbb{Z} 0$ for all $x:\mathbb{Z}$. By the computation rules for the integers type, we have

$$\beta_\mathbb{Z}^{\cdot, 0, 0}:0 \cdot 0 =_{\mathbb{Z}} 0$$

Then we need to show that for all $x:\mathbb{Z}$ we have an equivalence
$$(0 \cdot x =_{\mathbb{Z}} 0) \simeq (0 \cdot s(x) =_{\mathbb{Z}} 0)$$

But that's given by [[transport]] across the identification

$$\beta_\mathbb{Z}^{\cdot, 0, s}(x)^{-1}:0 \cdot x =_\mathbb{Z} 0 \cdot s(x)$$

for the type family $x =_\mathbb{Z} 0$:

$$\mathrm{transport}_{x:\mathbb{Z}.x =_\mathbb{Z} 0}(0 \cdot x, 0 \cdot s(x), \beta_\mathbb{Z}^{\cdot, 0, s}(x)^{-1}):(0 \cdot x =_{\mathbb{Z}} 0) \simeq (0 \cdot s(x) =_{\mathbb{Z}} 0)$$

Thus, by induction on the integers, we have

$$\mathrm{labsorp}_\mathbb{Z} \equiv \mathrm{ind}_\mathbb{Z}^{x:\mathbb{Z}.x \cdot 0 =_\mathbb{Z} 0}(\beta_\mathbb{Z}^{\cdot, 0, 0}, \lambda x:\mathbb{Z}.\mathrm{transport}_{x:\mathbb{Z}.x =_\mathbb{Z} 0}(0 \cdot x, 0 \cdot s(x), \beta_\mathbb{Z}^{\cdot, 0, s}(x)^{-1})):\prod_{x:\mathbb{Z}} 0 \cdot x =_\mathbb{Z} 0$$

Similarly, for the right absorption law, we have an equivalence
$$(x \cdot 0 =_{\mathbb{Z}} 0) \simeq (s(x) \cdot 0 =_{\mathbb{Z}} 0)$$

given by [[transport]] across the identification

$$\beta_\mathbb{Z}^{\cdot, s, 0}(x)^{-1}:x \cdot 0 =_\mathbb{Z} s(x) \cdot 0$$

for the type family $x =_\mathbb{Z} 0$:

$$\mathrm{transport}_{x:\mathbb{Z}.x =_\mathbb{Z} 0}(x \cdot 0, s(x) \cdot 0, \beta_\mathbb{Z}^{\cdot, s, 0}(x)^{-1}):(x \cdot 0 =_{\mathbb{Z}} 0) \simeq (s(x) \cdot 0 =_{\mathbb{Z}} 0)$$

Thus, by induction on the integers, we have

$$\mathrm{rabsorp}_\mathbb{Z} \equiv \mathrm{ind}_\mathbb{Z}^{x:\mathbb{Z}.0 \cdot x =_\mathbb{Z} 0}(\beta_\mathbb{Z}^{\cdot, 0, 0}, \lambda x:\mathbb{Z}.\mathrm{transport}_{x:\mathbb{Z}.x =_\mathbb{Z} 0}(x \cdot 0, s(x) \cdot 0, \beta_\mathbb{Z}^{\cdot, s, 0}(x)^{-1})):\prod_{x:\mathbb{Z}} x \cdot 0 =_\mathbb{Z} 0$$

Thus, the integers are an absorption magma with respect to zero and multiplication.
\end{proof}

\begin{theorem}
Multiplication in the integers type is bilinear with respect to addition.
\end{theorem}

\begin{proof}
We begin with bilinearity in the right parameter of multiplication, and we prove by double induction on the integers: given an integer $x:\mathbb{Z}$, we need to show the following types have elements:

$$x \cdot (0 + 0) =_\mathbb{Z} x \cdot 0 + x \cdot 0$$
$$\prod_{y:\mathbb{Z}} (x \cdot (0 + y) =_\mathbb{Z} x \cdot 0 + x \cdot y) \simeq (x \cdot (0 + s(y)) =_\mathbb{Z} x \cdot 0 + x \cdot s(y))$$
$$\prod_{y:\mathbb{Z}} (x \cdot (y + 0) =_\mathbb{Z} x \cdot y + x \cdot 0) \simeq (x \cdot (s(y) + 0) =_\mathbb{Z} x \cdot s(y) + x \cdot 0)$$
$$\prod_{y:\mathbb{Z}} \prod_{z:\mathbb{Z}} (x \cdot (y + z) =_\mathbb{Z} x \cdot y + x \cdot z) \simeq (x \cdot (s(y) + s(z)) =_\mathbb{Z} x \cdot s(y) + x \cdot s(z))$$

Now, for the first type, by application of the function $\lambda z:\mathbb{Z}.x \cdot z$ on the identification $\beta_\mathbb{Z}^{+, 0, 0}:0 + 0 =_\mathbb{Z} 0$, we have

$$\mathrm{ap}_{\lambda z:\mathbb{Z}.x \cdot z}(0 + 0, 0, \beta_\mathbb{Z}^{+, 0, 0}):x \cdot (0 + 0) =_\mathbb{Z} x \cdot 0$$

Similarly, by the right absorption law of multiplication, we have

$$\mathrm{rabsorp}_\mathbb{Z}(x):x \cdot 0 =_\mathbb{Z} 0$$

and by [[function application to identifications|application of the function]] $\lambda z:\mathbb{Z}.x \cdot 0 + z$ across $\mathrm{rabsorp}_\mathbb{Z}(x)$, we have

$$\mathrm{ap}_{\lambda z:\mathbb{Z}.x \cdot 0 + z}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)):x \cdot 0 + x \cdot 0 =_\mathbb{Z} x \cdot 0 + 0$$

By the right unit law for addition, we have

$$\mathrm{runitadd}_{\mathbb{Z}}(x \cdot 0):x \cdot 0 + 0 =_{\mathbb{Z}} x \cdot 0$$

Thus, by concatenating identifications we have

$$\mathrm{rdist}_\mathbb{Z}^{0, 0}(x) \equiv \mathrm{ap}_{\lambda z:\mathbb{Z}.x \cdot z}(0 + 0, 0, \beta_\mathbb{Z}^{+, 0, 0}) \bullet \mathrm{runitadd}_{\mathbb{Z}}(x \cdot 0)^{-1} \bullet \mathrm{ap}_{\lambda z:\mathbb{Z}.x \cdot 0 + z}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x))^{-1}:x \cdot (0 + 0) =_\mathbb{Z} x \cdot 0 + x \cdot 0$$

Next, we need to show that for all $y:\mathbb{Z}$ there is an equivalence of types

$$(x \cdot (0 + y) =_\mathbb{Z} x \cdot 0 + x \cdot y) \simeq (x \cdot (0 + s(y)) =_\mathbb{Z} x \cdot 0 + x \cdot s(y))$$

By [[transport]] across the identification $\mathrm{rabsorp}_\mathbb{Z}(x):x \cdot 0 =_\mathbb{Z} 0$ for type families

$$x \cdot (0 + y) =_\mathbb{Z} z + x \cdot y$$
and
$$x \cdot (0 + s(y)) =_\mathbb{Z} z + x \cdot s(y)$$

we get equivalences

$$\mathrm{transport}_{z:\mathbb{Z}.x \cdot (0 + y) =_\mathbb{Z} z + x \cdot y}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)):(x \cdot (0 + y) =_\mathbb{Z} x \cdot 0 + x \cdot y) \simeq (x \cdot (0 + y) =_\mathbb{Z} 0 + x \cdot y)$$

and

$$\mathrm{transport}_{z:\mathbb{Z}.x \cdot (0 + s(y)) =_\mathbb{Z} z + x \cdot s(y)}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)):(x \cdot (0 + s(y)) =_\mathbb{Z} x \cdot 0 + x \cdot s(y)) \simeq (x \cdot (0 + y) =_\mathbb{Z} 0 + x \cdot s(y))$$

Similarly, by [[transport]] across the identification
$$\mathrm{lunitadd}_\mathbb{Z}(x \cdot y):0 + x \cdot y =_\mathbb{Z} x \cdot y$$
for type family
$$x \cdot (0 + y) =_\mathbb{Z} z$$
we get the equivalence
$$\mathrm{transport}_{z:\mathbb{Z}.x \cdot (0 + y) =_\mathbb{Z} z}(0 + x \cdot y, x \cdot y, \mathrm{lunitadd}_\mathbb{Z}(x \cdot y)):(x \cdot (0 + y) =_\mathbb{Z} 0 + x \cdot y) \simeq (x \cdot (0 + y) =_\mathbb{Z} x \cdot y)$$
and by [[transport]] across the identification
$$\mathrm{lunitadd}_\mathbb{Z}(x \cdot s(y)):0 + x \cdot s(y) =_\mathbb{Z} x \cdot s(y)$$
for type family
$$x \cdot (0 + s(y)) =_\mathbb{Z} z$$
we get the equivalence
$$\mathrm{transport}_{z:\mathbb{Z}.x \cdot (0 + s(y)) =_\mathbb{Z} z}(0 + x \cdot s(y), x \cdot s(y), \mathrm{lunitadd}_\mathbb{Z}(x \cdot s(y))):(x \cdot (0 + s(y)) =_\mathbb{Z} 0 + x \cdot s(y)) \simeq (x \cdot (0 + s(y)) =_\mathbb{Z} x \cdot s(y))$$

By transport across the identification
$$\mathrm{lunitadd}_\mathbb{Z}(y):0 + y =_\mathbb{Z} y$$
for type family
$$x \cdot z =_\mathbb{Z} x \cdot y$$
we get the equivalence
$$\mathrm{transport}_{z:\mathbb{Z}.x \cdot z =_\mathbb{Z} x \cdot y}(0 + y, y, \mathrm{lunitadd}_\mathbb{Z}(y)):(x \cdot (0 + y) =_\mathbb{Z} x \cdot y) \simeq (x \cdot y =_\mathbb{Z} x \cdot y)$$
and by transport across the identification
$$\mathrm{lunitadd}_\mathbb{Z}(s(y)):0 + s(y) =_\mathbb{Z} s(y)$$
for type family
$$x \cdot z =_\mathbb{Z} x \cdot s(y)$$
we get the equivalence
$$\mathrm{transport}_{z:\mathbb{Z}.x \cdot z =_\mathbb{Z} x \cdot y}(0 + s(y), s(y), \mathrm{lunitadd}_\mathbb{Z}(y)):(x \cdot (0 + s(y)) =_\mathbb{Z} x \cdot s(y)) \simeq (x \cdot s(y) =_\mathbb{Z} x \cdot s(y))$$

Thus, it remains to show that there is an equivalence
$$(x \cdot y =_\mathbb{Z} x \cdot y) \simeq (x \cdot s(y) =_\mathbb{Z} x \cdot s(y))$$
But since $\mathbb{Z}$ is a set, both identity types are propositions, and have elements $\mathrm{refl}_\mathbb{Z}(x \cdot y)$ and $\mathrm{refl}_\mathbb{Z}(x \cdot s(y))$ respectively. As a result, both identity types are [[contractible types]], and contractible types are equivalent to each other. Thus, we have an equivalence:
$$f(x, y):(x \cdot y =_\mathbb{Z} x \cdot y) \simeq (x \cdot s(y) =_\mathbb{Z} x \cdot s(y))$$

Finally, by composing all the equivalences above, we get an equivalence

$$\begin{array}{c}
\mathrm{rdist}_\mathbb{Z}^{0, s}(x, y) \equiv \mathrm{transport}_{z:\mathbb{Z}.x \cdot (0 + s(y)) =_\mathbb{Z} z + x \cdot s(y)}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)) \circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot (0 + s(y)) =_\mathbb{Z} z}(0 + x \cdot s(y), x \cdot s(y), \mathrm{lunitadd}_\mathbb{Z}(x \cdot s(y)))\\
\circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot z =_\mathbb{Z} x \cdot y}(0 + s(y), s(y), \mathrm{lunitadd}_\mathbb{Z}(s(y))) \circ f(x, y) \\
\circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot z =_\mathbb{Z} x \cdot y}(0 + y, y, \mathrm{lunitadd}_\mathbb{Z}(y)) \circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot (0 + y) =_\mathbb{Z} z}(0 + x \cdot y, x \cdot y, \mathrm{lunitadd}_\mathbb{Z}(x \cdot y)) \\
\circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot (0 + y) =_\mathbb{Z} z + x \cdot y}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)):(x \cdot (0 + y) =_\mathbb{Z} x \cdot 0 + x \cdot y) \simeq (x \cdot (0 + s(y)) =_\mathbb{Z} x \cdot 0 + x \cdot s(y))
\end{array}$$

By a similar argument, we have

$$\begin{array}{c}
\mathrm{rdist}_\mathbb{Z}^{s, 0}(x, y) \equiv \mathrm{transport}_{z:\mathbb{Z}.x \cdot (s(y) + 0) =_\mathbb{Z} x \cdot s(y) + z}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)) \circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot (s(y) + 0) =_\mathbb{Z} z}(x \cdot s(y) + 0, x \cdot s(y), \mathrm{runitadd}_\mathbb{Z}(x \cdot s(y)))\\
\circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot z =_\mathbb{Z} x \cdot y}(s(y) + 0, s(y), \mathrm{runitadd}_\mathbb{Z}(s(y))) \circ f(x, y) \\
\circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot z =_\mathbb{Z} x \cdot y}(y + 0, y, \mathrm{runitadd}_\mathbb{Z}(y)) \circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot (y + 0) =_\mathbb{Z} z}(x \cdot y + 0, x \cdot y, \mathrm{runitadd}_\mathbb{Z}(x \cdot y)) \\
\circ \mathrm{transport}_{z:\mathbb{Z}.x \cdot (y + 0) =_\mathbb{Z} x \cdot y + z}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)):(x \cdot (y + 0) =_\mathbb{Z} x \cdot y + x \cdot 0) \simeq (x \cdot (s(y) = 0) =_\mathbb{Z} x \cdot s(y) + x \cdot 0)
\end{array}$$

Finally, we need to show that for all $y:\mathbb{Z}$ and $z:\mathbb{Z}$ there is an equivalence of types

$$(x \cdot (y + z) =_\mathbb{Z} x \cdot y + x \cdot z) \simeq (x \cdot (s(y) + s(z)) =_\mathbb{Z} x \cdot s(y) + x \cdot s(z))$$

...
\end{proof}

\begin{definition}
The integer number **one** $1:\mathbb{Z}$ is defined as

$$1 \equiv s(0):\mathbb{Z}$$
\end{definition}

\begin{theorem}
The integers type is a [[unital magma]] with respect to the one integer $1:\mathbb{Z}$ and multiplication $x:\mathbb{Z}, y:\mathbb{Z} \vdash x \cdot y:\mathbb{Z}$ both defined above.
\end{theorem}

\begin{proof}
We begin with the right unital laws, that $x \cdot s(0) =_\mathbb{Z} x$. By the computation rules for the integers type, we have

$$\beta_\mathbb{Z}^{\cdot, 0, s}(0):0 \cdot s(0) =_{\mathbb{Z}} 0 \cdot 0$$

Concatenating with $\beta_\mathbb{Z}^{\cdot, 0, 0}$ results in the required identification

$$\beta_\mathbb{Z}^{\cdot, 0, s}(0) \bullet \beta_\mathbb{Z}^{\cdot, 0, 0}:0 \cdot s(0) =_{\mathbb{Z}} 0$$

Then we need to show that for all $x:\mathbb{Z}$ we have an equivalence
$$(x \cdot s(0) =_{\mathbb{Z}} x) \simeq (s(x) \cdot s(0) =_{\mathbb{Z}} s(x))$$

We have the family of identifications
$$x:\mathbb{Z} \vdash \beta_\mathbb{Z}^{\cdot, s, s}(x, 0):s(x) \cdot s(0) =_{\mathbb{Z}} x \cdot 0 + s(x + 0)$$

Since multiplication is an absorption magma, we have identifications
$$\mathrm{rabsorp}_\mathbb{Z}(x):x \cdot 0 =_\mathbb{Z} 0$$
so [[function application to identifications|application of the function]] $\lambda z:\mathbb{Z}.z + s(x + 0)$ to the above identification results in the identification
$$\mathrm{ap}_{\lambda z:\mathbb{Z}.z + s(x + 0)}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)):x \cdot 0 + s(x + 0) =_{\mathbb{Z}} 0 + s(x + 0)$$
The left unit law for addition results in
$$\mathrm{lunitadd}_\mathbb{Z}(s(x + 0)):0 + s(x + 0) =_{\mathbb{Z}} s(x + 0)$$
and application of the successor equivalence to the right unit law results in
$$\mathrm{ap}_s(x + 0, x, \mathrm{runitadd}_\mathbb{Z}(x)):s(x + 0) =_{\mathbb{Z}} s(x)$$

Concatenating all the identifications leads to the identification

$$\beta_\mathbb{Z}^{\cdot, s, s}(x, 0) \bullet \mathrm{ap}_{\lambda z:\mathbb{Z}.z + s(x + 0)}(x \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(x)) \bullet \mathrm{lunitadd}_\mathbb{Z}(s(x + 0)) \bullet \mathrm{ap}_s(x + 0, x, \mathrm{runitadd}_\mathbb{Z}(x)):s(x) \cdot s(0) =_{\mathbb{Z}} s(x)$$

Similarly, by substituting $s^{-1}(x)$ in for $x$ in the above expression, we have

$$\beta_\mathbb{Z}^{\cdot, s, s}(s^{-1}(x), 0) \bullet \mathrm{ap}_{\lambda z:\mathbb{Z}.z + s(s^{-1}(x) + 0)}(s^{-1}(x) \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(s^{-1}(x))) \bullet \mathrm{lunitadd}_\mathbb{Z}(s(s^{-1}(x) + 0)) \bullet \mathrm{ap}_s(s^{-1}(x) + 0, s^{-1}(x), \mathrm{runitadd}_\mathbb{Z}(s^{-1}(x))):s(s^{-1}(x)) \cdot s(0) =_{\mathbb{Z}} s(s^{-1}(x))$$

In addition, we have the right inverse [[homotopy]]
$$\mathrm{rinv}_s:\prod_{x:\mathbb{Z}} s(s^{-1}(x)) =_\mathbb{Z} x$$
so by [[transport]] across the identity $\mathrm{rinv}_s(x)$ for the type family $z \cdot s(0) =_\mathbb{Z} x$, we also have the equivalence
$$\mathrm{transport}_{z:\mathbb{Z}.z \cdot s(0) =_\mathbb{Z} x}(s(s^{-1}(x)), x, \mathrm{rinv}_s(x)):(s(s^{-1}(x)) \cdot s(0) =_{\mathbb{Z}} x) \simeq (x \cdot s(0) =_{\mathbb{Z}} x)$$

Thus, we have the identification
$$\begin{array}{c}
\mathrm{transport}_{z:\mathbb{Z}.z \cdot s(0) =_\mathbb{Z} x}(s(s^{-1}(x)), x, \mathrm{rinv}_s(x))(\beta_\mathbb{Z}^{\cdot, s, s}(s^{-1}(x), 0) \bullet \mathrm{ap}_{\lambda z:\mathbb{Z}.z + s(s^{-1}(x) + 0)}(s^{-1}(x) \cdot 0, 0, \mathrm{rabsorp}_\mathbb{Z}(s^{-1}(x))) \\
\bullet \mathrm{lunitadd}_\mathbb{Z}(s(s^{-1}(x) + 0)) \bullet \mathrm{ap}_s(s^{-1}(x) + 0, s^{-1}(x), \mathrm{runitadd}_\mathbb{Z}(s^{-1}(x))) \bullet \mathrm{rinv}_s(x)):x \cdot s(0) =_{\mathbb{Z}} x
\end{array}$$

Since $\mathbb{Z}$ is a set, the [[identity types]] $x \cdot s(0) =_{\mathbb{Z}} x$ and $s(x) \cdot s(0) =_{\mathbb{Z}} s(x)$ are both propositions, and since they have identifications, both types are [[contractible types]]. All contractible types are equivalent to each other; thus there is an equivalence

$$f(x):(x \cdot s(0) =_{\mathbb{Z}} x) \simeq (s(x) \cdot s(0) =_{\mathbb{Z}} s(x))$$

Then, by induction on the integers, there is the homotopy

$$\mathrm{runitmul}_\mathbb{Z}:\mathrm{ind}_{\mathbb{Z}}^{x \cdot s(0) =_{\mathbb{Z}} x}(\beta_\mathbb{Z}^{\cdot, 0, s}(0) \bullet \beta_\mathbb{Z}^{\cdot, 0, 0}, \lambda x:\mathbb{Z}.f(x)):\prod_{x:\mathbb{Z}} x \cdot s(0) =_{\mathbb{Z}} x$$

Next, we derive the left unital laws, that $s(0) \cdot x =_\mathbb{Z} x$. By the computation rules for the integers type, we have

$$\beta_\mathbb{Z}^{\cdot, s, 0}(0):s(0) \cdot 0 =_{\mathbb{Z}} 0 \cdot 0$$

Concatenating with $\beta_\mathbb{Z}^{\cdot, 0, 0}$ results in the required identification

$$\beta_\mathbb{Z}^{\cdot, s, 0}(0) \bullet \beta_\mathbb{Z}^{\cdot, 0, 0}:s(0) \cdot 0 =_{\mathbb{Z}} 0$$

Then we need to show that for all $x:\mathbb{Z}$ we have an equivalence
$$(s(0) \cdot x =_{\mathbb{Z}} x) \simeq (s(0) \cdot s(x) =_{\mathbb{Z}} s(x))$$

We have the family of identifications
$$x:\mathbb{Z} \vdash \beta_\mathbb{Z}^{\cdot, s, s}(0, x):s(0) \cdot s(x) =_{\mathbb{Z}} 0 \cdot x + s(0 + x)$$

Since multiplication is an absorption magma, we have identifications
$$\mathrm{labsorp}_\mathbb{Z}(x):0 \cdot x =_\mathbb{Z} 0$$
so [[function application to identifications|application of the function]] $\lambda z:\mathbb{Z}.z + s(0 + x)$ to the above identification results in the identification
$$\mathrm{ap}_{\lambda z:\mathbb{Z}.z + s(0 + x)}(0 \cdot x, 0, \mathrm{labsorp}_\mathbb{Z}(x)):0 \cdot x + s(0 + x) =_{\mathbb{Z}} 0 + s(0 + x)$$
The left unit law for addition results in
$$\mathrm{lunitadd}_\mathbb{Z}(s(0 + x)):0 + s(0 + x) =_{\mathbb{Z}} s(0 + x)$$
and application of the successor equivalence to the left unit law results in
$$\mathrm{ap}_s(0 + x, x, \mathrm{runitadd}_\mathbb{Z}(x)):s(0 + x) =_{\mathbb{Z}} s(x)$$

Concatenating all the identifications leads to the identification

$$\beta_\mathbb{Z}^{\cdot, s, s}(0, x) \bullet \mathrm{ap}_{\lambda z:\mathbb{Z}.z + s(0 + x)}(0 \cdot x, 0, \mathrm{labsorp}_\mathbb{Z}(x)) \bullet \mathrm{lunitadd}_\mathbb{Z}(s(0 + x)) \bullet \mathrm{ap}_s(0 + x, x, \mathrm{lunitadd}_\mathbb{Z}(x)):s(0) \cdot s(x) =_{\mathbb{Z}} s(x)$$

Similarly, by substituting $s^{-1}(x)$ in for $x$ in the above expression, we have

$$\beta_\mathbb{Z}^{\cdot, s, s}(0, s^{-1}(x)) \bullet \mathrm{ap}_{\lambda z:\mathbb{Z}.z + s(0 + s^{-1}(x))}(0 \cdot s^{-1}(x), 0, \mathrm{labsorp}_\mathbb{Z}(s^{-1}(x))) \bullet \mathrm{lunitadd}_\mathbb{Z}(s(0 + s^{-1}(x))) \bullet \mathrm{ap}_s(0 + s^{-1}(x), s^{-1}(x), \mathrm{lunitadd}_\mathbb{Z}(s^{-1}(x))):s(0) \cdot s(s^{-1}(x)) =_{\mathbb{Z}} s(s^{-1}(x))$$

In addition, we have the right inverse [[homotopy]]
$$\mathrm{rinv}_s:\prod_{x:\mathbb{Z}} s(s^{-1}(x)) =_\mathbb{Z} x$$
so by [[transport]] across the identity $\mathrm{rinv}_s(x)$ for the type family $s(0) \cdot z =_\mathbb{Z} x$, we also have the equivalence
$$\mathrm{transport}_{z:\mathbb{Z}.s(0) \cdot z =_\mathbb{Z} x}(s(s^{-1}(x)), x, \mathrm{rinv}_s(x)):s(0) \cdot (s(s^{-1}(x)) =_{\mathbb{Z}} x) \simeq (s(0) \cdot x =_{\mathbb{Z}} x)$$

Thus, we have the identification
$$\begin{array}{c}
\mathrm{transport}_{z:\mathbb{Z}.s(0) \cdot z =_\mathbb{Z} x}(s(s^{-1}(x)), x, \mathrm{rinv}_s(x))
(\beta_\mathbb{Z}^{\cdot, s, s}(0, s^{-1}(x)) \bullet \mathrm{ap}_{\lambda z:\mathbb{Z}.z + s(0 + s^{-1}(x))}(0 \cdot s^{-1}(x), 0, \mathrm{labsorp}_\mathbb{Z}(s^{-1}(x))) \\
\bullet \mathrm{lunitadd}_\mathbb{Z}(s(0 + s^{-1}(x))) \bullet \mathrm{ap}_s(0 + s^{-1}(x), s^{-1}(x), \mathrm{lunitadd}_\mathbb{Z}(s^{-1}(x))) \bullet \mathrm{rinv}_s(x)):s(0) \cdot x =_{\mathbb{Z}} x
\end{array}$$

Since $\mathbb{Z}$ is a set, the [[identity types]] $s(0) \cdot x =_{\mathbb{Z}} x$ and $s(0) \cdot s(x) =_{\mathbb{Z}} s(x)$ are both propositions, and since they have identifications, both types are [[contractible types]]. All contractible types are equivalent to each other; thus there is an equivalence

$$g(x):(s(0) \cdot x =_{\mathbb{Z}} x) \simeq (s(0) \cdot s(x) =_{\mathbb{Z}} s(x))$$

Then, by induction on the integers, there is the homotopy

$$\mathrm{lunitmul}_\mathbb{Z}:\mathrm{ind}_{\mathbb{Z}}^{s(0) \cdot x =_{\mathbb{Z}} x}(\beta_\mathbb{Z}^{\cdot, s, 0}(0) \bullet \beta_\mathbb{Z}^{\cdot, 0, 0}, \lambda x:\mathbb{Z}.g(x)):\prod_{x:\mathbb{Z}} s(0) \cdot x =_{\mathbb{Z}} x$$

Thus, $\mathbb{Z}$ is a [[unital magma]] with respect to one $1 \equiv s(0):\mathbb{Z}$ and multiplication $x:\mathbb{Z}, y:\mathbb{Z} \vdash x \cdot y:\mathbb{Z}$.
\end{proof}

\begin{theorem}
Multiplication on the integers is associative.
\end{theorem}

\begin{proof}
....
\end{proof}

\begin{theorem}
Multiplication on the integers is commutative.
\end{theorem}

\begin{proof}
....
\end{proof}

\begin{corollary}
The integers are a [[commutative ring]].
\end{corollary}

### Exponentiation on the integers

+--{: .num_defn}
###### Definition
The binary operation **exponentiation** is defined by induction on the natural numbers

Introduction rules for exponentiation:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N}, x:\mathbb{Z}, \vdash x^n:\mathbb{Z}}$$

Computation rules for exponentiation:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{Z}, \vdash \beta_\mathbb{Z}^{x^{(-)}, 0}(x):x^0 =_\mathbb{Z} 1}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N},  x:\mathbb{Z},  \vdash \beta_\mathbb{Z}^{x^{(-)}, s}:x^{s(n)} =_\mathbb{Z} x^n \cdot x}$$
=--

+--{: .num_defn}
###### Definition
The inclusion of the natural numbers in the integers is inductively defined by induction on the natural numbers

Introduction rules for natural number inclusion:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash i(n):\mathbb{Z}}$$

Computation rules for natural number inclusion:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \beta_\mathbb{Z}^{i, 0}:i(0) =_\mathbb{Z} 0}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \beta_\mathbb{Z}^{i, s}(n):i(s(n)) =_\mathbb{Z} s(i(n))}$$
=--

### Decidability of equality

Perhaps it is time to prove that equality is decidable on the integers.

+--{: .num_defn}
###### Definition
The **observational equality** relation is defined by double induction on the integers

Formation rules for observational equality:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{Z}, y:\mathbb{Z} \vdash \mathrm{Eq}_\mathbb{Z}(x, y) \; \mathrm{type}}$$

Definition rules for observational equality:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \beta_\mathbb{Z}^{\mathrm{Eq}_\mathbb{Z}, 0, 0}:\mathrm{Eq}_\mathbb{Z}(0, 0) \simeq \mathbb{1}}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{Z} \vdash \beta_\mathbb{Z}^{\mathrm{Eq}_\mathbb{Z}, 0, s}:\mathrm{Eq}_\mathbb{Z}(0, s(x)) \simeq \mathrm{Eq}_\mathbb{Z}(s^{-1}(0), x)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{Z} \vdash \beta_\mathbb{Z}^{\mathrm{Eq}_\mathbb{Z}, 0, s}:\mathrm{Eq}_\mathbb{Z}(s(x), 0) \simeq \mathrm{Eq}_\mathbb{Z}(x, s^{-1}(0))}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{Z}, y:\mathbb{Z} \vdash \beta_\mathbb{Z}^{\mathrm{Eq}_\mathbb{Z}, s, s}:\mathrm{Eq}_\mathbb{Z}(s(x), s(y)) \simeq \mathrm{Eq}_\mathbb{Z}(x, y)}$$
=--

The following rules only hold if the universe has a [[univalent universe]]:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, N:\mathbb{N} \vdash \beta_\mathbb{Z}^{\mathrm{Eq}_\mathbb{Z}, s, 0}:\mathrm{Eq}_\mathbb{Z}(0, i(n) + 1) \simeq \mathbb{0}}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, N:\mathbb{N} \vdash \beta_\mathbb{Z}^{\mathrm{Eq}_\mathbb{Z}, 0, s}:\mathrm{Eq}_\mathbb{Z}(i(n) + 1, 0) \simeq \mathbb{0}}$$

### Order structure on the integers ###

Given a type family $x:A \vdash B(x)$, we define the [[existential quantifier]] $\exists x:A.B(x)$ as the [[propositional truncation]] of the [[dependent sum type]] $\left[\sum_{x:A} B(x)\right]$. Given two types $A$ and $B$, we define the [[disjunction]] $A \vee B$ as the [[propositional truncation]] of the [[sum type]] $\left[A + B\right]$.

+--{: .num_defn}
###### Definition
The predicate *is less than*, denoted as $x \lt y$, is defined as

$$x \lt y \coloneqq \exists n:\mathbb{N}.x + i(n) + 1 =_\mathbb{N} y$$

for $x:\mathbb{Z}$, $y:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is greater than*, denoted as $x \gt y$, is defined as

$$x \gt y \coloneqq \exists n:\mathbb{N}.x =_\mathbb{N} y + i(n) + 1$$

for $x:\mathbb{Z}$, $y:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is apart from*, denoted as $x # y$, is defined as

$$x # y \coloneqq (x \lt y) \vee (x \gt y)$$

for $x:\mathbb{Z}$, $y:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is less than or equal to*, denoted as $x \leq y$, is defined as

$$x \leq y \coloneqq \exists n:\mathbb{N}.x + i(n) =_\mathbb{N} y$$

for $x:\mathbb{Z}$, $y:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is greater than or equal to*, denoted as $x \geq y$, is defined as

$$x \geq y \coloneqq \exists n:\mathbb{N}.x =_\mathbb{N} y + i(n)$$

for $x:\mathbb{Z}$, $y:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is positive*, denoted as $\mathrm{isPositive}(x)$, is defined as

$$\mathrm{isPositive}(x) \coloneqq \exists n:\mathbb{N}.i(n) + 1 =_\mathbb{N} x$$

for $x:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is negative*, denoted as $\mathrm{isNegative}(x)$, is defined as

$$\mathrm{isNegative}(x) \coloneqq \exists n:\mathbb{N}.x + i(n) + 1 =_\mathbb{N} 0$$

for $x:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is zero*, denoted as $\mathrm{isZero}(x)$, is defined as

$$\mathrm{isZero}(x) \coloneqq x =_\mathbb{Z} 0$$

for $x:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is non-positive*, denoted as $\mathrm{isNonPositive}(x)$, is defined as

$$\mathrm{isNonPositive}(x) \coloneqq \exists n:\mathbb{N}.0 =_\mathbb{N} x + i(n)$$

for $x:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is non-negative*, denoted as $\mathrm{isNonNegative}(x)$, is defined as

$$\mathrm{isNonNegative}(x) \coloneqq \exists n:\mathbb{N}.x =_\mathbb{N} i(n)$$

for $x:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The predicate *is non-zero*, denoted as $\mathrm{isNonZero}(x)$, is defined as

$$\mathrm{isNonZero}(x) \coloneqq \mathrm{isPositive}(x) \vee \mathrm{isNegative}(x)$$

for $x:\mathbb{Z}$.
=--

### Pseudolattice and metric structure on the integers ###

+--{: .num_defn}
###### Definition
The **ramp function** $\mathrm{ramp}:\mathbb{Z} \to \mathbb{Z}$ is defined as

$$\mathrm{ramp}(i(n)) \coloneqq i(n)$$

$$\mathrm{ramp}(-(i(n) + 1)) \coloneqq 0$$

for $n:\mathbb{N}$.
=--

+--{: .num_defn}
###### Definition
The **minimum** $\min:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is defined as

$$\min(x,y) \coloneqq x - \mathrm{ramp}(x - y)$$

for $x:\mathbb{Z}$, $y:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The **maximum** $\max:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is defined as

$$\max(x,y) \coloneqq x + \mathrm{ramp}(y - x)$$

for $x:\mathbb{Z}$, $y:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The **metric** $\rho:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is defined as

$$\rho(x,y) \coloneqq \max(x,y) - \min(x,y)$$

for $x:\mathbb{Z}$, $y:\mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The **absolute value** $\vert(-)\vert:\mathbb{Z} \to \mathbb{Z}$ is defined as

$$\vert x \vert \coloneqq \max(x, -x)$$

for $x:\mathbb{Z}$.
=--

### Division and remainder ###

+--{: .num_defn}
###### Definition
**Integer division** $(-)\div(-):\mathbb{Z} \times \mathbb{Z}_{\neq 0} \to \mathbb{Z}$ is defined as

$$i(n) \div i(m) \coloneqq i(n \div m)$$
$$i(n) \div -i(m) \coloneqq -i(n \div m)$$
$$-i(n) \div i(m) \coloneqq -i(n \div m)$$
$$-i(n) \div -i(m) \coloneqq i(n \div m)$$

for $n:\mathbb{N}$, $m:\mathbb{N}_{\neq 0}$.
=--

## Categorical semantics

The categorical semantics of the integers type is the [[integers object]].

## See also

* [[integers object]]

* [[natural numbers type]]

* [[equivalence type]]

* [[line type]]

* [[free group type]]

## References

For definitions as a [[higher inductive type]] in [[homotopy type theory]], see:

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

* Thorsten Altinkirch, _The Integers as a Higher Inductive Type_, ([abs:2007.00167](https://arxiv.org/abs/2007.00167))

* Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson, _Symmetry_, ([pdf](https://unimath.github.io/SymmetryBook/book.pdf))

A few more definitions of the integers, as well as the statement of the dependent universal property of the integers, could be found in:

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

Some terms about specific kinds of [[H-spaces]] used to prove properties of the integers are defined in:

* {#BuchholtzEtAl2023} [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

A formalization in terms of [[homotopy type theory]], using a unary notation, is in

* [[Mike Shulman]], _[Integers.v](https://github.com/HoTT/HoTT/blob/master/Coq/HIT/Integers.v)_

(A different common formalization of integers in type theory is in a binary notation, as in the [Coq standard library](http://coq.inria.fr/stdlib/Coq.ZArith.BinInt.html).  Binary notation is exponentially more efficient for performing computations, but the unary notation was convenient for calculating $\pi_1(S^1)$.)

That one can construct the [[natural numbers type]] from the integers type can be found in:

* [[Christian Sattler]], *Natural numbers from integers* ([pdf](https://www.cse.chalmers.se/~sattler/docs/naturals.pdf))

[[!redirects integers type]]
[[!redirects integers types]]

[[!redirects universal property of the integers type]]
[[!redirects dependent universal property of the integers type]]

[[!redirects universal property of the integers]]
[[!redirects dependent universal property of the integers]]


integrability of G-structures

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Given a [[G-structure]], it is _integrable_ or _locally flat_ if over [[germs]] it restricts to the canonical (trivial) $G$-structure. If it restricts to the canonical structure only over order-$k$ [[infinitesimal disks]], then it is called _order $k$ integrable_. The [[obstruction]] to first-order integrability is the [[torsion of a G-structure]], and hence first-order integrable $G$-structures are also called _torsion-free $G$-structures_.

Beware that some authors use the term "integrable" for "torsion-free". This originates in concentration on the case of [[almost symplectic structure]], i.e. $G$-structure for $G = Sp(2n)$ the [[symplectic group]], in which case the [[Darboux theorem]] gives that first order integrability (to [[symplectic structure]]) already implies full integrability. However, in general this is not the case. For instance for [[orthogonal structure]], i.e. $G$-structure for $G = O(n)$ the [[orthogonal group]], then the [[fundamental theorem of Riemannian geometry]] gives that the torsion obstruction to first-order integrability vanishes, exhibited by the [[Levi-Civita connection]], but full integrability here is equivalent to this being a [[flat connection]], which is a strong additional constraint. This is the case from which the terminology "locally flat" for "integrable" derives from.

## Definition
 {#Definition}

### Traditional
 {#DefinitionTraditional}

Let $V$ be a linear local model space, e.g. a [[vector space]] in plain [[differential geometry]] or [[super vector space]] in [[supergeometry]], etc.. Write $GL(V)$ for its [[general linear group]]. Consider a [[group]] [[homomorphism]] $G \longrightarrow GL(V)$.

Write $\mathbf{c}_0$ for the _standard flat $G$-structure_ on $V$ (see at _[G-Structure -- Examples -- Standard flat G-structure](G-structure#TheStandardFlatGStructure)_).

+-- {: .num_defn}
###### Definition

A [[G-structure]] $\mathbf{c}$ on a [[manifold]] $X$ modeled on $V$ (e.g. a [[smooth manifold]] or [[supermanifold]]) is called integrable if

1. there exists [[cover]] $\{U_i \hookrightarrow X\}$ by [[open subsets]]  $U_i \hookrightarrow V$;

1. such that the $G$-structure $\mathbf{c}$ on $X$ restricts on each patch to the default $G$-structure $\mathbf{c}_0$ on $V$:

   $$
     \mathbf{c}|_{U_i} \simeq \mathbf{c}_0|_{U_i}
     \,.
   $$

=--

This is due to ([Sternberg 64, section VII, def. 2.4](#Sternberg64), [Guillemin 65, section 3](#Guillemin65)).  For review see also ([Alekseevskii](#Alekseevskii), [Lott 90, page 4 of the exposition](#Lott90)).

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

More concretely, if $G$-structure is modeled by $G$-subbundles $P$ of the [[frame bundle]] (as discussed at _[G-structure -- In terms of subbundles of the frame bundle](G-structure#InTermsOfSubbundlesOfTheFrameBundle)_ ), then it is integrable if each $P \hookrightarrow Fr(X)$ restricts on each patch to $P_0 \hookrightarrow Fr(V)$

$$
  \array{
     P_0|_{U_i} &\hookrightarrow& Fr(V)|_{U_i}
     \\
     {}^{\mathllap{\simeq}}\downarrow && \downarrow
     \\
     P|_{U_i} &\hookrightarrow& Fr(X)|_{U_i}
  }
  \,.
$$

=--

+-- {: .num_defn}
###### Definition

For $k \in \mathbb{N}$, a [[G-structure]] $\mathbf{c}$ on a [[manifold]] $X$ modeled on $V$ (e.g. a [[smooth manifold]] or [[supermanifold]]) is called _order-$k$ infinitesimally integrable_ if at each point $x \in X$ its restriction to the order-$k$ [[infinitesimal neighbourhood]] $\mathbb{D}^V_0 \simeq \mathbb{D}_x^X \hookrightarrow X$ is equal to the default $G$-structure $\mathbf{c}_0$.

=--

([Guillemin 65, section 4](#Guillemin65))

### In higher differential cohesive geometry
 {#IntermOfDifferentialCohesion}

One may formalize the concept of integrable $G$-structure in the generality of [[higher differential geometry]], formalized in [[differential cohesion]].  See also  there at _[differential cohesion -- G-Structure](differential+cohesive+%28infinity%2C1%29-topos#structures)_.

Let $V$ be framed, def. \ref{Framing}, let $G$ be an [[∞-group]] and $G \to GL(V)$ a homomorphism, hence

$$
  G\mathbf{Struc}\colon \mathbf{B}G \longrightarrow \mathbf{B}GL(V)
$$

a morphism between the [[deloopings]].

+-- {: .num_defn #GStructure}
###### Definition

For $X$ a $V$-manifold, def. \ref{Manifold}, a **[[G-structure]]** on $X$ is a lift of the structure group of its [[frame bundle]], def. \ref{FrameBundleMap}, to $G$, hence a diagram

$$
  \array{
     X && \longrightarrow&&  \mathbf{B}G
     \\
     & {}_{\mathllap{\tau_X}}\searrow && \swarrow_{\mathrlap{G\mathbf{Struc}}}
     \\
     && \mathbf{B}GL(V)
  }
$$

hence a morphism

$$
  \mathbf{c} \colon \tau_X \longrightarrow G\mathbf{Struc}
$$

is the [[slice (∞,1)-topos]].

=--

In fact $G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(n)}$ is the [[moduli ∞-stack]] of such $G$-structures.

The double [[slice (∞,1)-topos|slice]] $(\mathbf{H}_{/\mathbf{B}GL(n)})_{/G\mathbf{Struc}}$ is the [[(∞,1)-category]] of such $G$-structures.

+-- {: .num_example #TrivialGStructure}
###### Example

If $V$ is framed, def. \ref{Framing}, then it carries the trivial $G$-structure, which we denote by

$$
  \mathbf{c}_0 \colon \tau_{V} \longrightarrow G\mathbf{Struc}
  \,.
$$

=--

+-- {: .num_defn #IntegrableGStructure}
###### Definition

For $V$ framed, def. \ref{Framing}, and $X$ a $V$-manifold, def. \ref{Manifold}, then  $G$-structure $\mathbf{c}$ on $X$ is _[[integrability of G-structure|integrable]]_ (or _locally flat_) if there exists a $V$-cover

$$
  \array{
     && U
     \\
     & \swarrow && \searrow
     \\
     V && && X
  }
$$

such that the [[correspondence]] of [[frame bundles]] induced via remark \ref{FrameBundlesFunctorial}

$$
  \array{
     && \tau_U
     \\
     & \swarrow && \searrow
     \\
     \tau_V && && \tau_X
  }
$$

(a diagram in $\mathbf{H}_{/\mathbf{B}GL(V)}$) extends to a sliced correspondence between $\mathbf{c}$ and the trivial $G$-structure $\mathbf{c}_0$ on $V$, example \ref{TrivialGStructure}, hence to a diagram in $\mathbf{H}_{/\mathbf{B}GL(V)}$ of the form

$$
  \array{
     && \tau_U
     \\
     & \swarrow && \searrow
     \\
     \tau_V && \swArrow_{\mathrlap{\simeq}}  && \tau_X
     \\
     & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && G\mathbf{Struct}
  }
$$

On the other hand, $\mathbf{c}$ is called _infinitesimally integrable_ (or _torsion-free_) if such an extension exists (only) after restriction to all [[infinitesimal disks]] in $X$ and $U$, hence after composition with the [[counit of a comonad|counit]]

$$
  \flat^{rel} U \longrightarrow U
$$

of the [[relative flat modality]], def. \ref{InducedRelativeShapeAndFlat}:

$$
  \array{
     && \tau_{\flat^{rel} U}
     \\
     & \swarrow && \searrow
     \\
     \tau_V && \swArrow_{\mathrlap{\simeq}}  && \tau_X
     \\
     & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && G\mathbf{Struct}
  }
 \,.
$$

=--

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

As before, if the given [[reduction modality]] encodes order-$k$ infinitesimals, then the infinitesimal integrability in def. \ref{IntegrableGStructure} is order-$k$ integrability. For $k = 1$ this is [[torsion of a G-structure|torsion-freeness]].

=--

## Properties

### Existence and torsion

The [[obstruction]] for a $G$-structure to be integrable to first order is its [[torsion of a G-structure]].

### In terms of adapted coordinate systems

A $G$-structure on $X$ is integrable previsely if there exists an [[atlas]] of $X$ by [[coordinate charts]] with the property that their canonical [[frame fields]] are $G$-frames.

([Sternberg 64, section VII, exercise 2.1](#Sternberg64))

## Examples
 {#Examples}

* [Complex structure](#ExampleComplexStructure)

* [Symplectic structire](#ExampleSymplecticStructure)

* [Orthogonal structure](#ExamplesOrthogonalStructure)

* [G2-structure](#ExampleG2Structure)

### Complex structure
 {#ExampleComplexStructure}

A $GL(n,\mathbb{C}) \to GL(2n,\mathbb{R})$-[[G-structure|structure]] is an _[[almost complex structure]]_. Its [[torsion of a G-structure]] vanishes precisely if its [[Nijenhuis tensor]] vanishes, hence, by the [[Newlander-Nirenberg theorem]], precisely if it is a [[complex structure]]. Since a [[complex manifold]] admits holomorphic [[coordinate charts]], this first-order integrability already implies full integrability.

### Symplectic structure
 {#ExampleSymplecticStructure}

An $Sp(n) \hookrightarrow GL(2n)$-[[G-structure|structure]] is an _[[almost symplectic structure]]_. Its [[torsion of a G-structure]] is the [[de Rham differential]] $\mathbf{d}\omega$ of the corresponding 2-form $\omega$ (recalled e.g. in [Albuquerque-Picken 11](symplectic+manifold#AlbuquerquePicken11)). Hence first-order integrability here amounts precisely to [[symplectic structure]]. The [[Darboux theorem]] asserts that this is already a fully integrable structure.

### Orthogonal structure
 {#ExamplesOrthogonalStructure}

An $O(n)\to GL(n)$-[[G-structure|structure]] is an [[orthogonal structure]], hence a [[vielbein]], hence a [[Riemannian metric]]. The [[fundamental theorem of Riemannian geometry]] says that in this case the [[torsion of a G-structure]] vanishes, exhibited by the existence of the [[Levi-Civita connection]]. The corresponding first-order integrability is the existence of [[Riemann normal coordinates]] (since these identify the given [[vielbein]] at any point to first order with the trivial (identity) vielbein). The higher order obstructions to integrability turn out to all be proportional to combinations of the [[Riemann curvature]]. Full integrability is equivalent to the vanishing of Riemann tensor, hence to the LC-connection being a [[flat connection]].

### Unitary structure
 {#UnitaryStructure}

The case of unitary structure is precisely the combination of the above three cases.

By the fact (see at _[unitary group -- relation to orthogonal, symplectic and general linear group](unitary+group#RelationToOrthogonalSymplecticAndGeneralLinearGroup)_) that the [[unitary group]] is the intersection

$$
  U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})
$$

a $U(n) \hookrightarrow GL(2n,\mathbb{R})$-structure -- called an _[[almost Hermitian structure]]_ -- is precisely a joint [[orthogonal structure]], [[almost symplectic structure]] and [[almost complex structure]]. Hence if first order integrable -- called a [[Kähler manifold]] structure -- this is precisely a joint [[orthogonal structure]]/[[Riemannian manifold]] structure, [[symplectic manifold]] structure, [[complex manifold]] structure.

### $G_2$-Structure
 {#ExampleG2Structure}

A An $G_2 \to GL(7)$-[[G-structure|structure]] is a [[G2-structure]]. Its [[torsion of a G-structure]] vanishes if the corresponding definite 3-form $\omega$ is [[covariant derivative|covariantly constant]] with respect to the induced [[Riemannian metric]], in which case the structure is a [[G2-manifold]]. Beware that some authors refer to first-order integrable $G_2$-structure (or even weaker conditions) as "integrable $G_2$-structure" (see [Bryant 05, remark 2](G2+manifold#Bryant05) for critical discussion of the terminology). The  [[torsion of a G-structure|higher-order torsion invariants]] of $G_2$-structures do not in general vanish (e.g [Bryant 05, (4.7)](G2+manifold#Bryant05)) and so, contrary to the above cases of symplectic and complex structure, $G_2$-manifold structure does not imply integrable $G_2$-structure.

### Further examples

* [[CR-manifold]]

* [[torsion constraints of supergravity]]

## References

* {#Sternberg64} [[Shlomo Sternberg]], chapter VII of _Lectures on differential geometry_, Prentice-Hall (1964)

* {#Guillemin65} [[Victor Guillemin]], _The integrability problem for $G$-structures_, Trans. Amer. Math. Soc. 116 (1965), 544–560. ([jstor:1994134](https://www.jstor.org/stable/1994134))

* {#Alekseevskii} D. V: Alekseevskii, _$G$-structure on a manifold_ in M. Hazewinkel (ed.) _Encyclopedia of Mathematics, Volume 4_ ([eom:G-structure](https://encyclopediaofmath.org/wiki/G-structure))

Lecture notes include

* [[Marius Crainic]], _[Differential geometry course](https://webspace.science.uu.nl/~crain101/DG-2015/)_, 2015 ([pdf](https://webspace.science.uu.nl/~crain101/DG-2015/main10.pdf), [[CrainicDifferentialGeometry15.pdf:file]])

* {#Pasquotto} [[Federica Pasquotto]], _Linear $G$-structures by examples_ ([pdf](http://www.few.vu.nl/~pasquott/course16.pdf), [[PasquottoGStructures.pdf:file]])

Discussion with an eye towards [[torsion constraints in supergravity]] is in

* {#Lott90} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_, Comm. Math. Phys. 133 (1990), 563–615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125))

Discussion with an eye towards [[special holonomy]] is in

* {#Joyce00} [[Dominic Joyce]], _Compact manifolds with special holonomy_, Oxford University Press 2000

Discussion in [[modal homotopy type theory]]:

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017

* {#Wellen18} [[Felix Cherubini]] (née Wellen), _Cartan Geometry in Modal Homotopy Type Theory_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

See also the references at _[[torsion of a Cartan connection]]_ and at _[[torsion constraints in supergravity]]_.

[[!redirects integrability of G-structure]]
[[!redirects integrability of G-structures]]

[[!redirects integrable G-structure]]
[[!redirects integrable G-structures]]

[[!redirects integrable G-stucture]]
[[!redirects integrable G-stuctures]]

[[!redirects first-order integrable G-structure]]
[[!redirects first-order integrable G-structures]]


integrable differential equation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Roughly, "integrating" a [[partial differential equation]] means to find a [[solution]] for it, usually understood with special properties, such as having prescribed boundary data (initial data), and/or having "closed form", such as expression by [[quadratures]] ("[[integrable systems]]") and/or having prescribed [[domain]] (e.g. over [[infinitesimal neighbourhoods]] in [[formal integrability]]).  A [[PDE]] is "integrable" in a prescribed sense if it admits solutions in this prescribed sense.

## Related concepts

* [[formally integrable PDE]]

* [[integrable system]]

## References

* {#Goldschmidt67} [[Hubert Goldschmidt]], _Integrability criteria for systems of nonlinear partial differential equations_, Journal of Differential Geometry 1 (1967) 269–307 ([Euclid](https://projecteuclid.org/euclid.jdg/1214428094))

* [[Maciej Zworski]], _Numerical linear algebra and solvability of partial differential equations_, Communications in Mathematical Physics 229 (2002) 293–307

[[!redirects integrable differential equations]]

[[!redirects integrable partial differential equation]]
[[!redirects integrable partial differential equations]]

[[!redirects integrable PDE]]
[[!redirects integrable PDEs]]


integrable distribution of subspaces

> This entry is about the concept of subspaces of [[vector bundles]] and [[Lie algebroids]]. For the concept in functional analysis see at _[[distribution]]_.

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## Idea

Let $p:V\to M$ be a [[smooth vector bundle]]. Any smooth family of $k$-dimensional subspaces $W_m\subset p^{-1}(m)$ where $m\in M$ is called a _[[distribution of subspaces|distribution of k-dimensional subspaces]]_ in $V\to M$. If $V = T M$ is the [[tangent bundle]] of $M$ then we talk about distributions of tangent vectors.

A distribution of [[tangent vectors]] is called _integrable_ if the [[Lie bracket]] of its [[sections]] is **involutive**, i.e. if $X,Y: M\to W\subset T M$ are two sections (vector fields belonging to the distribution) of $W\to M$ then the bracket $[X,Y]$ of these vector fields is also a section of $W$: $[X,Y]\in W$.

More generally, if a vector bundle is equipped with the structure of a [[Lie algebroid]], then a [[distribution of subspaces]] is _integrable_ if its sections are closed under the given Lie bracket. This reduces to the previous case for the [[tangent Lie algebroid]]. Hence *integrable distributions are [[subobject|sub-]]Lie algebroids*.

## Properties

A basic result on integrability is the Frobenius theorem ([Wikipedia](http://en.wikipedia.org/wiki/Frobenius_theorem_%28differential_topology%29)) which relates involutivity to integrability in the sense of partial differential equations. Examples include complex analytic manifolds which correspond exactly to complex manifolds with an integrable almost complex structure. [[Courant algebroid]]s are a quite general tool to express the integrability of geometric structure that include these as special cases.

## Related concepts

* [[foliation]]

  * [[foliation of a Lie algebroid]]

  * [[foliation of a Lie groupoid]]

* [[polarization]]

* [[exterior differential system]]

[[!redirects integrable distributions]]

[[!redirects integrable distribution]]
[[!redirects integrable distributions]]


integrable system

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##Idea##

Models of theoretical [[physics]] are often given in the form of [[differential equations]]. Informally, [[integrable PDE|integrability]] is the property of a concrete [[model (physics)|model]] which enables one to [[solution|solve]] these equations in a closed form, or in terms of [[quadratures]] (expressions in terms of ordinary [[integrals]]).

##Classical integrable systems##

In [[classical mechanics]], a sort of integrability follows if there are generalized "angle-action" coordinates; the evolution of a system is then of a very special form, hence such "[[Liouville integrability]]" (existence of a maximal [[Poisson manifold|Poisson]]-commuting invariants) leads to a special type of global structure of orbits. In addition to finite-dimensional integrable models, many classical PDEs of mathematical physics can be exactly solved, which can often be expressed in terms of infinite-dimensional Hamiltonian systems. There are many methods and approaches which are often simultaneously applicable, and under some assumptions essentially equivalent; we list just keywords: Lax pairs, bihamiltonian systems, spectral curve, classical r-matrix, inverse scattering method, [[Riemann-Hilbert problem|Riemann--Hilbert method]] and Birkhoff decomposition (which also provide connections to the subject of special functions)...

Much of the knowledge about the algebras of differential operators on curves has been obtained from research on Calogero--Moser systems and generalizations (and their connections to the KP hierarchy). An important class of integrable systems are special cases of [[Hitchin fibration|Hitchin system]].

## Quantum integrability

Often classically integrable systems after quantization may be also exactly solved, or show some other form of integrability, but there are no general rules when this is possible. Quantum integrability usually means that we can find a complete system of commuting operators whose common eigenstates will form a basis of the [[Hilbert space]] of the model. Many examples are found making manipulations with algebras of operators; as a consequence generalizations of existing models often involve combinatorial data with algebaric meaning; say for example generalized Calogero models related to the root systems. Quantum R-matrices and [[quantum group|quantum groups]]
were invented with a view toward quantum integrable systems systematizing analogues of a classical quantum integrability trick called the Bethe Ansatz.

## Related entries

* [[Liouville integrability]],

* [[integrable PDE]]

* [[soliton]]

* [[spin chain]], [[Toda lattice]], [[Bethe ansatz]]

* [[sine-Gordon model]]

* [[quasitriangular bialgebra]], [[Yangian]], [[quantum group]]

* [[Lax equation]], [[spectral curve]]

* [[Burchnall-Chaundy theory]], [[argument shift method]]

* [[tau-function]], [[Sato Grassmannian]]

* [[Calogero model]], [[Dunkl operator]]

* [[topological recursion]]

## References

### General

* O. Babelon, D. Bernard, M. Talon, _Introduction to classical integrable systems_, Cambridge Univ. Press 2003.
* V.E. Korepin, N. M. Bogoliubov, A. G. Izergin, _Quantum inverse scattering method and correlation functions_, Cambridge Univ. Press 1997.
* Ron Donagi, Eyal Markman, _Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles_, pp. 1-119 in Integrable systems and quantum groups (expanded CIME lectures, Montecatini Terme, 1993), Lecture Notes in Math. __1620__ (1996) [arXiv:alg-geom/9507017](http://arxiv.org/abs/alg-geom/9507017), [MR97h:14017](http://www.ams.org/mathscinet-getitem?mr=1397273)
* Eyal Markman, _Spectral curves and integrable systems_ Compositio Math. __93__ no. 3 (1994), p. 255-290, [numdam](http://www.numdam.org/item?id=CM_1994__93_3_255_0)
* See [wikipedia](http://en.wikipedia.org/wiki/Integrable_system) for an overview and a list of some famous integrable systems and their respective pages.
* M.R. Adams, J. Harnad, J. Hurtubise, _Integrable Hamiltonian systems on rational coadjoint orbits of loop algebras, Hamiltonian systems, transformation groups and spectral transform methods_, Proc. CRM Workshop, Montreal 1989 (1990) pp. 19–32
* [[Michèle Audin]], _Hamiltonian systems and their integrability_, SMF/AMS Texts and Monographs, vol. __15__ (2008)
* A.Levin, M.Olshanetsky, A.Smirnov, A.Zotov, _Characteristic classes and integrable systems. General construction_, [arxiv/1006.0702](https://arxiv.org/abs/1006.0702)
* Yuji Kodama, Lauren Williams, _KP solitons, total positivity, and cluster algebras_, Proc. Natl. Acad. Sci. [arxiv/1105.4170](https://arxiv.org/abs/1105.4170), [doi](doi:10.1073/pnas.1102627108)
* Pierre Van Moerbeke, [[David Mumford]], _The spectrum of difference operators and algebraic curves_, Acta Math. __143__, n. 1, 93-154, [MR80e:58028](https://mathscinet.ams.org/mathscinet-getitem?mr=533894), [doi](https://doi.org/10.1007/BF02392090)
* [[Boris Dubrovin|B. A. Dubrovin]], _Completely integrable Hamiltonian systems that are associated with matrix operators, and Abelian varieties_, Funkcional. Anal. i Priložen. __11__ (1977), no. 4, 28-41, 96, [MR0650114](http://www.ams.org/mathscinet-getitem?mr=0650114)
* [[Boris Dubrovin|B. A. Dubrovin]], _Integrable systems and Riemann surfaces lecture notes_, [pdf](http://people.sissa.it/~dubrovin/rsnleq_web.pdf)
* [[Kevin Costello]], [[Edward Witten]], Masahito Yamazaki, _Gauge theory and integrability I_, [arxiv:1709.09993](https://arxiv.org/abs/1709.09993)

Discussion in the context of [[harmonic maps]] is in

* [[Shabnam Beheshti]], A. Shadi Tahvildar-Zadeh, _Integrability and vesture for harmonic maps into symmetric spaces_ ([arXiv:1209.1383](https://arxiv.org/abs/1209.1383))

See also

* Wikipedia, _[Integrable system](https://en.wikipedia.org/wiki/Integrable_system)_

### In string/M-theory

On integrability in [[Seiberg-Witten theory]]:

* [[Alexander Gorsky]], [[Igor Krichever]], [[Andrei Marshakov]], [[Andrei Mironov]], [[Andrey Morozov]], *Integrability and Seiberg-Witten Exact Solution*, Phys. Lett. B **355** (1995) 466-474 [[arXiv:hep-th/9505035](https://arxiv.org/abs/hep-th/9505035), doi:10.1016/0370-2693(95)00723-X]

* [[Andrei Marshakov]], *Seiberg-Witten Curves and Integrable Systems* [[arXiv:hep-th/9903252](https://arxiv.org/abs/hep-th/9903252)]

On integrability in [[topological string theory]]:

* [[Mina Aganagic]], [[Robbert Dijkgraaf]], [[Albrecht Klemm]], [[Marcos Marino]], [[Cumrun Vafa]], pp. 7 of: *Topological Strings and Integrable Hierarchies*, Commun. Math. Phys. **261** (2006) 451-516 [[arXiv:hep-th/0312085](https://arxiv.org/abs/hep-th/0312085), [doi:10.1007/s00220-005-1448-9](https://doi.org/10.1007/s00220-005-1448-9)]

* [[Mina Aganagic]], [[Miranda Cheng]], [[Robbert Dijkgraaf]], Daniel Krefl, [[Cumrun Vafa]], §2 in: *Quantum Geometry of Refined Topological Strings*, J. High Energ. Phys. **2012** 19 (2012) [[arXiv:1105.0630](https://arxiv.org/abs/1105.0630), doi:10.1007/JHEP11(2012)019]

On possible structures in [[M2-brane]] dynamics and [[M2-M5-brane bound states]] which could be [[M-theory]]-lifts of the familiar [[integrable system|integrability]] of the [[Green-Schwarz superstring]] on [[AdS spacetime|$AdS_5$]] $\times$ [[5-sphere|$S^5$]]:

* [[Kirill Gubarev]], [[Edvard Musaev]], *Integrability structures in string theory* [[arXiv:2301.06486](https://arxiv.org/abs/2301.06486)]

[[!redirects integrable systems]]

[[!redirects integrable model]]
[[!redirects integrable models]]

[[!redirects integrability]]

[[!redirects integrable hierarchy]]
[[!redirects integrable hierarchies]]


integral

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## Idea

_Integration_ is a process by which local data over a [[manifold]] or similar is _accumulated_ to an _integral_.

In its simplest form, this is a limiting notion of the process of forming [[sums]], known then as _[[Riemann integration]]_ or _[[Lebesgue integration]]_, where the integration pairs a [[function]] on a space against a _[[measure]]_ which indicates, roughly, how much a local contribution of the function contributes to the whole accumulation process.
Direct variants and refinements of this kind of integration is _[[integration of differential forms]]_ and similar.

The integration of differential forms induces a more general notion of integration, namely [[integration in differential cohomology]] and hence [[integration in generalized cohomology]]. Here the choice of a [[measure]] is replaced by a choice of _[[orientation in generalized cohomology]]_.

| analytic integration | cohomological integration |
|--|--|
| [[measure]] | [[orientation in generalized cohomology]] |
| [[Lebesgue integration]], [[integration of differential forms|of differential forms]] | [[push-forward in generalized cohomology]]/[[fiber integration in differential cohomology|in differential cohomology]] |

## Terminology

The quantity whose integral we are taking is the __integrand__.  Depending on context, we may view the integrand as a [[function]], a [[differential form]], a [[measure]], or otherwise.  The result of integrating the integrand is the __value__ of the integral.  The __integral__ itself may be thought of as its value, but also (especially if this value fails to exist as we would like) could mean the entire expression.  (Contrast this with the case of [[series]], where 'series' means the entire expression but 'sum' means the result of evaluating it.)  Thus, in $\int_a^b f(x) \,\mathrm{d}x$, any of $f$, $f(x)$, $f(x) \,\mathrm{d}x$, with or without the additional data of the set $[a,b]$ or the proposition $a \leq x \leq b$, may be thought of as the integrand.  The integral is $\int_a^b f(x) \,\mathrm{d}x$, which may be thought of as consisting of all of the data in this expression or as simply being the final result of evaluating the expression.

An integral of the type considered here is sometimes called a __definite integral__ to contrast with an [[indefinite integral|indefinite]] (or semidefinite) integral.  Whereas a definite integral is typically a number or some other definite quantity, an indefinite integral is typically another variable quantity of the same type as the integrand (or even more indefinitely, an equivalence class of such quantities).  Thus, $\int_a^b f(x) \,\mathrm{d}x$ is a definite integral, $\int_a f(x) \,\mathrm{d}x$ is a semidefinite integral, and $\int f(x) \,\mathrm{d}x$ is an indefinite integral.

__Integration__ in the most narrow sense is a process involved in defining or computing integrals.

## Analytic integration
 {#AnalyticIntegration}

### Generalizing sums
 {#GeneralizingSums}

A __[[sum]]__ $\sum_{s \in S} a_s$ is defined over a domain $S$ which is, as a rule, a _discrete_ set. This set is also typically fixed in the sense that no subdivision is assumed in its definition, except sometimes for convergence purposes. Thus a general sum takes two arguments: a set $S$ and a function on $S$ and outputs a value. In the case of sum of an infinite series the order may matter as the limiting procedure is taken into account.

An __integral__ is a generalization of a sum where the domain is typically not a discrete set, but some mathematical object, which may be (typically not in a single fixed way, but in many possible ways) approximated or split into pieces.
The function to be integrated over the range,
is some object which is thought of
as living over the range, some sort of a cocycle or a distribution.
The basic property from the integral is that it should behave nicely (typically additively in some sense) with respect to combining its values on pieces in the range and that, for a reasonable set of subdivisions the result should be the same (sometimes after passing to a limit).

For a fixed range, integral is typically an operator/functional on a set/space of allowed objects over the range.

### Integration as opposite to differentiation

Many integrals are supposed to be inverse to differentiation procedures of various kinds. Indeed, if integral is a generalization of a sum, then a difference between two partial sums is a value to be added at a step of summation, and its generalization is some sort of [[differentiation]]. Of course, the initial value has to be determined as the differentiation gives just a step of the addition.

### Solving differential equations and constraints

In some cases, one solves a differential equation by reducing it to a relation of the form $d F = g$ and then $F = \int g$. One says that the equation is
solvable in quadratures. Thus the integration is used in examples of solving differential equations and differential relations, hence finding objects satisfying some differential constraints is often also considered a sort of integration. For example, finding integral curves of vector fields and more generally finding integral submanifolds of distributions, is also called an integration. In this vain, also a Lie group is a global object which integrates a Lie algebra (indeed, infinitesimally this reduces to solving the Maurer-Cartan equations). For resolving differential relations there are solvability conditions/obstructions/constraints which are often of cohomological nature. There is sometimes a relation to [[rational homotopy theory]].

## Topological integration
 {#TopologicaIntegration}

(...)

* [[integration in generalized cohomology]]
(...)

## Zoo of integrals and concepts of integration

We list a bunch more notions of integration. Should eventually be turned into something more coherent...

See also [[measure theory]] (and [[measurable space]], [[measure space]]) which is a basis for many kinds of integrals, especially the [[Lebesgue integral]].

Basic kinds of integrals in (super)analysis: the [[Daniell integral]], the [[Riemann integral]] (possibly [[improper Riemann integral|improper]] or [[Henstock integral|generalised]]), the [[Lebesgue integral]], the (upper or lower) [[Darboux integral]], the [[Henstock integral]] (same as the generalised Riemann integral or the improper Lebesgue integral), the [[Stieltjes integral|Stieltjes]] variations of the above, the [[Berezin integral]], [[line integrals]].

Special examples of the above include the [[Batalin-Vilkovisky integral]], the [[Kontsevich integral]], the [[Selberg integral]],the [[elliptic Selberg integral]].

Integration is involved in [[integral transform | integral transforms]], [[integral transforms on sheaves]], in various formulas for pairings, e.g. of chains and cochains ....
Some statements involving integrals include the [[Stokes theorem]].

A special topic includes some infinite-dimensional versions including the well-defined [[Wiener integral]] and the more problematic [[path integral]], cf. also

* [[fermionic path integral]],
* [[path integral as a pull-push transform]].

The basic problem with the path integral comes from the fact that there is no translation-invariant [[Lebesgue measure]] on an infinite-dimensional real vector space with a finite nonzero value on the unit ball.

* [[fiber integration]]

* in [[differential geometry]]

  *  [[differential forms]] are involved in integration over differentiable manifolds, see at _[[integration of differential forms]]_ ; more generally, one can integrate currents (cf. [[geometric measure theory]]

  * [[fiber integration in ordinary differential cohomology]]

*  in [[supergeometry]]

   * [[Berezin integral]]

   * [[integration over supermanifolds]]
* [[path integral]]
* [[Lie integration]] (the name comes from its relation to the integration of
differential equations and finding integral curves of
vector fields and flows)

* stratified versions related to Grothendieck rings and valuations:
[[Euler integration]], [[motivic integration]]

## Properties

* [[Cauchy integral theorem]]

(...)

## Related concepts

* [[integrand]]

* [[closed midpoint algebra]]

* [[Lebesgue integration]]

* [[Lebesgue space]]

* [[integration of differential forms]]

* [[change of integration variables]]

* [[density]], [[volume form]]

* [[integration over infinite-dimensional manifolds]]

* [[fiber integration]], [[fiber integration in differential cohomology]]

* [[Gaussian integral]], [[Feynman diagram]]

* [[Lie integration]]

* [[geometric integration theory]]

* [[cohomological integration]]

* [[adelic integration]]

[[!include infinitesimal and local - table]]

## References

An introduction to the basic notions of [[integration of differential forms]], with an eye towards applications in [[physics]] is in section 3 of

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

Discussion of integration similarly with an eye towards applications in physics but from a more [[general abstract]] perspective is in

* _[[geometry of physics]] -- [integration](geometry+of+physics#Integration)_

A proof of the [[Riesz representation theorem]] in [[constructive mathematics]] is given in

* [[Thierry Coquand]], [[Bas Spitters]], _Integrals and Valuations_, Logic and Analysis (2009) 1(3) p.1-22, [arXiv:0808.1522](http://arxiv.org/abs/0808.1522), [doi](http://dx.doi.org/10.4115/jla.2009.1.3)

For the use of [[category theory]] in [[integration]], see

* [[Igor Kriz]], [[Ales Pultr]], *Categorical Geometry and Integration Without Points*, Applied Categorical Structures (2014) Vol. 22, p.79-97 ([arXiv:1101.3762](https://arxiv.org/abs/1101.3762), [doi:10.1007/s10485-012-9295-2](https://doi.org/10.1007/s10485-012-9295-2))

For [[integration]] in [[constructive mathematics]], see

* [[Bas Spitters]], *Constructive algebraic integration theory*, Annals of Pure and Applied Logic, Volume 137, Issues 1–3, January 2006, Pages 380-390 ([pdf](https://www.cs.au.dk/~spitters/ftop2.pdf), [doi:10.1016/j.apal.2005.05.031](https://doi.org/10.1016/j.apal.2005.05.031))

* [[Bas Spitters]], *Constructive algebraic integration theory without choice*. In [[Thierry Coquand]], [[Henri Lombardi]], [[Marie-Francoise Roy]], Mathematics, Algorithms, Proofs, 2005, Dagstuhl Seminar Proceedings 05021, Internationales Begegnungs- und Forschungszentrum (IBFI), Schloss Dagstuhl, Germany. ([pdf](https://www.cs.au.dk/~spitters/obs.pdf))

category: analysis

[[!redirects integral]]
[[!redirects integrals]]
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[[!redirects integrations]]

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[[!redirects definite integrals]]
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integral calculus

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## Idea

Integral calculus is one of the two halves of the [[infinitesimal number|infinitesimal]] [[calculus]], the other being [[differential calculus]]. (The two are linked by the [[fundamental theorem of calculus]].)

Integral calculus describes [[integration]] of sufficiently well-behaved [[functions]].


integral closure

\tableofcontents

## Definition

Given a commutative ring $k$ and a field $L\supset k$, an element $x\in L$ is said to be __integral__ over $k$ if it satisfies a [[monic polynomial]] equation with coefficients in $k$, or equivalently, there exist a finitely-generated nonzero $k$-submodule $M\subset L$ such that $x M \subset M$.

A commutative ring $K\supset k$ is said to be __integral__ over $k$ if every element of $K$ is integral over $k$. The relation of integrality of overrings is transitive. If $f:K\to K'$ is a surjective homomorphism of rings and $K$ integral over $k\subset K$, then $K' = f(K)$ is integral over $f(k)$.

The set of all elements of $L$ integral over $k$ is a subring of $L$ called the __integral closure__ of $k$ in $L$. We say that $k$ is __integrally closed in__ $L$ if it equals its own integral closure in $L$.

An [[integral domain]] $k$ is __integrally closed__ if it is integrally closed in the quotient field of $k$.

## Properties

If $k$ is an integrally closed [[Noetherian ring|Noetherian domain]] and $L$ a finite separable field extension of its quotient field $Q(k)$ then the integral closure of $k$ in $L$ is finitely generated over $k$.

If $k$ is a principal ideal ring and $L$ a finite separable extension of degree $n$ of its quotient field $Q(k)$, then the integral closure of $k$ in $L$ is a free rank $n$-module over $k$.

If $K$ is integral over a subring $k$ then for any multiplicative set $S\subset k$, the localization $S^{-1} K$ is integral over $S^{-1} k$.

Every [[unique factorization domain]] is integrally closed.

## In constructive mathematics

In constructive mathematics, integral closure and [[algebraic closure]] are not the same if the field is not a [[discrete field]]. Integral closure is usually better behaved than algebraic closure in fields which are not [[discrete fields]], because not every polynomial can be shown to have a well-defined degree, but every monic polynomial by definition has a well-defined degree. This is especially the case with the [[fundamental theorem of algebra]] for a set of [[complex numbers]], which in constructive mathematics states that the [[complex numbers]] are integrally closed, and is true for the [[modulated Cantor real numbers]] while not provable for the [[Dedekind real numbers]].

## See also

* [[fundamental theorem of algebra]]

## References

* Serge Lang, _Algebraic number theory_, GTM 110, Springer 1970, 2000
* O. Zariski, Samuel, _Commutative algebra_
* N. Bourbaki, _Commutative algebra_
* Z. Borevich, I. Shafarevich, _Number theory_
* E. Artin, J. Tate, _Class field theory_, 1967
* A. Weil, _Basic number theory_

[[!redirects integrally closed]]

[[!redirects integrally closed ring]]
[[!redirects integrally closed rings]]

[[!redirects integrally closed field]]
[[!redirects integrally closed fields]]


integral cohomology

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## Idea

Integral cohomology or "[[ordinary cohomology]]" (see there) is the ordinary version of [[Whitehead-generalized cohomology]], the one that is [[Brown representability theorem|represented]] by the [[Eilenberg-MacLane spectrum]] $H \mathbb{Z}$ with [[coefficients]] in the [[integers]].

Integral cohomology is best known maybe in its incarnation as [[singular cohomology]] or [[Čech cohomology]] with coefficients in the integers.

## Geometric models

* integral cohomology in degree 1 classifies [[complex line bundle]];

* integral cohomology in degree 2 classifies complex line [[bundle gerbe]] / line [[principal 2-bundle|2-bundle]]s;

* integral cohomology in degree $n$ classifies line [[circle n-bundles|n-bundles]].

## Related concepts

* [[rational cohomology]], [[real cohomology]], [[complex cohomology]]

## References

Discussion [[cohomology in homotopy type theory|in homotopy type theory]]:

* [[Guillaume Brunerie]], [[Axel Ljungström]], [[Anders Mörtberg]], *Synthetic Integral Cohomology in Cubical Agda*, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) **216** (2022) $[$[doi:10.4230/LIPIcs.CSL.2022.11](https://doi.org/10.4230/LIPIcs.CSL.2022.11)$]$

  > (in [[cubical type theory|cubical]] [[Agda]])


integral current

__Integral current__ is a measure theoretic generalization of a notion of a manifold with some compactness properties. It is a kind of [[current (distribution theory)]] with some additional conditions including certain integrality condition.

* Frank Morgan, _Geometric measure theory: a beginner's guide_
* page 22 in Urs Lang, _Introduction to geometric measure theory_, summer school notes, [pdf](https://people.math.ethz.ch/~lang/diab05_notes.pdf)


integral domain

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## Definition

Let $R$ be a [[commutative ring]], let $\mathrm{Reg}(R)$ be the [[monoid of regular elements]] in $R$, let $\neg \mathrm{Reg}(R)$ be the [[complement]] of $\mathrm{Reg}(R)$, the set of [[zero divisors]] in $R$, and let $I_{\neg \mathrm{Reg}(R)}$ be the [[two-sided ideal]] generated by $\neg \mathrm{Reg}(R)$. A commutative ring is *non-trivial* if $0 \neq 1$.

\begin{definition}
An **integral domain** is a non-trivial [[commutative ring]] $R$ such that every [[element]] in $I_{\neg \mathrm{Reg}(R)}$ is equal to zero, or that the [[quotient ring]] $R/I_{\neg \mathrm{Reg}(R)}$ is isomorphic to $R$.
\end{definition}

\begin{remark}
  Beware that some authors allow an integral domain to be [[non-commutative ring|non-commutative]].
\end{remark}

Since every element of the ideal generated by the set of [[zero divisors]] is equal to zero, and the integral domain is a non-trivial ring, which means that there is at least one zero divisor, it follows that zero is a zero divisor and if $a \cdot b = 0$, then $a = 0$ or $b = 0$. It then follows that every integral domain is a [[reduced ring]].

The [[ring of fractions]] of every integral domain is a [[field]], the [[field of fractions]].

The ring of [[formal power series]] of an integral domain is a [[local integral domain]].

While every [[ring homomorphism]] between [[fields]] is an [[injection]], it is not the case that every ring homomorphism between integral domains is an injection. The ring homomorphism which takes a [[local integral domain]] to its [[quotient ring|quotient]] by its [[Jacobson radical]] is an injection if and only if the local integral domain is already a field.

## Examples

* [[GCD domain]]
* [[Bezout domain]]
* [[unique factorization domain]]
* [[principal ideal domain]]
* [[Euclidean domain]]
* [[Dedekind domain]]
* [[field]]
* [[valuation ring]]

## In constructive mathematics

in constructive mathematics, there are different inequivalent ways to define an integral domain. The above definition is sometimes called a **ring without zero divisors** or a **weak integral domain**. Stronger notions include:

\begin{definition}
A **Heyting integral domain** is an integral domain which has a [[tight apartness relation]] and where every element apart from zero is [[regular element in a ring|regular]]. The [[ring of fractions]] of a Heyting integral domain is a [[Heyting field]].
\end{definition}

\begin{definition}
A **discrete integral domain** is a Heyting integral domain with decidable equality, or a Heyting integral domain where every element is either zero or regular. The [[ring of fractions]] of a discrete integral domain is a [[discrete field]].
\end{definition}

Rings without zero divisors and discrete integral domains are both definable in [[coherent logic]]. However, Heyting integral domains can only be defined in [[first-order logic]].

## Generalizations

Sometimes, one may want the trivial ring to be an integral domain, resulting in the notion of *[[possibly trivial integral domain]]*. In [LombardiQuitté2010](#LombardiQuitté2010), the authors define an integral domain to be a possibly trivial integral domain.

In principle, one could just as easily consider a [[commutative rig]] or [[commutative semiring]] $R$.  In that case, however, only the definition involving the cancellative property extends to rigs and semirings. Furthermore, we should add the additional requirement that addition in $R$ is cancellable (that is, addition by any element is injective), to make the analogue of the previous paragraph correct. Since these could also be done for general [[domains]] to turn them into what could be called [[domain rigs]] and [[domain semirings]], these could be called *commutative domain rigs* or *commutative domain semirings*.

## Related concepts

* [[domain (ring theory)]]

* [[commutative ring]]

* [[approximate integral domain]]

* [[possibly trivial integral domain]]

* [[field]], [[field of fractions]]

* [[torsion-free module]]

* [[integral monoid]]

* [[difference protoring]]

* [[reduced ring]]

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

## Reference

* {#LombardiQuitté2010} [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

See also

* Wikipedia, *[Integral domain](https://en.wikipedia.org/wiki/Integral_domain)*

On [[limits]] of [[integral domains]] among [[commutative rings]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Limit closures of classes of commutative rings*, Theory Appl. Categories **30** (2015) 229–304 [[tac:30-08](http://www.tac.mta.ca/tac/volumes/30/8/30-08abs.html), [pdf](https://www.math.mcgill.ca/barr/papers/domrefl.pdf)]

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[[!redirects discrete integral domains]]

[[!redirects Heyting integral domain]]
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integral form

By an *integral form* one many mean:

* in [[differential geometry]]:

  a [[differential form]] whose [[periods]] are [[integers]]; hence a closed differential form that is the image in [[de Rham cohomology]], under the [[de Rham isomorphism]], of a class in [[integral cohomology]].

* in [[supergeometry]]:

  a [[differential form on a supermanifold]] for which there is a notion of [[integration over supermanifolds]].

[[!redirects integral forms]]

category: disambiguation