# Homotopy Type Theory integers > history (changes)

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# Contents

## Idea

The integers as familiar from school mathematics.

## Definitions

The type of integers, denoted $\mathbb{Z}$, has several definitions as a higher inductive type.

### Definition 1

The integers are 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: $\nu_0: inj(0, 0) = inj(1, 0)$.

### Definition 2

The integers are defined as the higher inductive type generated by:

• A term $0 : \mathbb{Z}$.
• A function $s : \mathbb{Z} \to \mathbb{Z}$.
• A function $p_1 : \mathbb{Z} \to \mathbb{Z}$.
• A function $p_2 : \mathbb{Z} \to \mathbb{Z}$.
• A dependent product of identities representing that $p_1$ is a section of $s$:
$\sigma: \prod_{a:\mathbb{Z}} p_1(s(a)) = a$
• A dependent product of identities representing that $p_2$ is a retracion of $s$:
$\rho: \prod_{a:\mathbb{Z}} s(p_2(a)) = a$

### Definition 3

The integers are defined as the higher inductive type generated by:

• A term $0 : \mathbb{Z}$.
• A function $s : \mathbb{Z} \to \mathbb{Z}$.
• A function $p : \mathbb{Z} \to \mathbb{Z}$.
• A dependent product of identities representing that $p$ is a section of $s$:
$\sigma: \prod_{a:\mathbb{Z}} p(s(a)) = a$
• A dependent product of identities representing that $p$ is a retracion of $s$:
$\rho: \prod_{a:\mathbb{Z}} s(p(a)) = a$
• A dependent product of identities representing the coherence condition:
$\kappa: \prod_{a:\mathbb{Z}} ap_s(\sigma(a)) = \rho(a)$

### Definition 4

The integers are defined as the higher inductive type generated by:

• A term $0 : \mathbb{Z}$.
• A function $s : \mathbb{Z} \to \mathbb{Z}$.
• A function $n : \mathbb{Z} \to \mathbb{Z}$.
• An identity representing that zero and negative zero are equal: $\nu: n(0) = 0$.
• A dependent product of identities representing that negation is an involution:
$\iota: \prod_{a:\mathbb{Z}} n(n(a)) = a$
• A dependent product of identities representing the coherence condition for the above:
$\kappa_\iota: \prod_{a:\mathbb{Z}} ap_n(\iota(a)) = \iota(a)$
• A dependent product of identities representing that $n \circ s \circ n$ is a section of $s$:
$\sigma: \prod_{a:\mathbb{Z}} n(s(n(s(a)))) = a$
• A dependent product of identities representing that $n \circ s \circ n$ is a retracion of $s$:
$\rho: \prod_{a:\mathbb{Z}} s(n(s(n(a)))) = a$
• A dependent product of identities representing the coherence condition:
$\kappa: \prod_{a:\mathbb{Z}} ap_s(\sigma(a)) = \rho(a)$

### Definition 5

The integers are 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

TODO: Show that the integers are a ordered Heyting integral domain with decidable equality, decidable apartness, and decidable linear order, and that the integers are initial in the category of ordered Heyting integral domains.

We assume in this section that the integers are defined according to definition 1.

### Commutative ring structure on the integers

###### Definition

The integer zero $0:\mathbb{Z}$ is defined as

$0 \coloneqq inj(0, 0)$
###### Definition

Let $(-)+(-):\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ be addition of the natural numbers, let $\rho:\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ be the symmetric difference or metric of the natural numbers, and let $\lt:\mathbb{N} \times \mathbb{N} \to \mathbb{2}$ be the decidable strict order on the natural numbers into the booleans. The binary operation addition $(-)+(-):\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is inductively defined as

$inj(0,a) + inj(0,b) \coloneqq inj(0,a+b)$
$inj(1,a) + inj(1,b) \coloneqq inj(1,a+b)$
$inj(0,a) + inj(1,b) \coloneqq inj(a \lt b,\rho(a,b))$
$inj(1,a) + inj(0,b) \coloneqq inj(b \lt a,\rho(a,b))$

for $a:\mathbb{N}$, $b:\mathbb{N}$.

###### Definition

The unary operation negation $-(-):\mathbb{Z} \to \mathbb{Z}$ is defined as

$-inj(a, b) \coloneqq inj(\neg a, b)$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $\neg:\mathbf{2} \to \mathbf{2}$

###### Definition

The binary operation subtraction $(-)-(-):\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is defined as

$inj(a, b) - inj(c, d) \coloneqq inj(a, b) + inj(\neg c, d)$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$.

###### Definition

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

$1 \coloneqq inj(0,1)$
###### Definition

The binary operation multiplication $(-)\cdot(-):\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is defined as

$inj(a,b) \cdot inj(c,d) \coloneqq inj(a \oplus c,b \cdot d)$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$, $(-)\cdot(-):\mathbb{N} \times \mathbb{N} \to \mathbb{N}$, $(-)\oplus(-):\mathbf{2} \times \mathbf{2} \to \mathbf{2}$ (exclusive or binary operation in $\mathbf{2}$).

###### Definition

The right $\mathbb{N}$-action exponentiation $(-)^{(-)}:\mathbb{Z} \times \mathbb{N} \to \mathbb{Z}$ is inductively defined as

$inj(a,b)^0 \coloneqq inj(0,1)$
$inj(a,b)^{2n} \coloneqq inj(0,b^{2n})$
$inj(a,b)^{2n+1} \coloneqq inj(a,b^{2n+1})$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $n:\mathbb{N}$, $(-)^{(-)}:\mathbb{N} \times \mathbb{N} \to \mathbb{N}$.

### Order structure on the integers

###### Definition

The dependent type is positive, denoted as $isPositive(inj(a,b))$, is defined as

$isPositive(inj(a,b)) \coloneqq (a = 0) \times (b \gt 0)$

for $a:\mathbf{2}$, $b:\mathbb{N}$, dependent types $m\gt n$ indexed by $m, n:\mathbb{N}$.

###### Definition

The dependent type is negative, denoted as $isNegative(inj(a,b))$, is defined as

$isPositive(inj(a,b)) \coloneqq (a = 1) \times (b \gt 0)$

for $a:\mathbf{2}$, $b:\mathbb{N}$, dependent types $m\gt n$ indexed by $m, n:\mathbb{N}$.

###### Definition

The dependent type is zero, denoted as $isZero(inj(a,b))$, is defined as

$isZero(inj(a,b)) \coloneqq b = 0$

for $a:\mathbf{2}$, $b:\mathbb{N}$.

###### Definition

The dependent type is non-positive, denoted as $isNonPositive(inj(a,b))$ is defined as

$isNonPositive(inj(a,b)) \coloneqq isPositive(inj(a,b)) \to \emptyset$

for $a:\mathbf{2}$, $b:\mathbb{N}$.

###### Definition

The dependent type is non-negative, denoted as $isNonNegative(inj(a,b))$, is defined as

$isNonNegative(inj(a,b)) \coloneqq isNegative(inj(a,b)) \to \emptyset$

for $a:\mathbf{2}$, $b:\mathbb{N}$.

###### Definition

The dependent type is non-zero, denoted as $isNonZero(inj(a,b))$, is defined as

$isNonZero(inj(a,b)) \coloneqq \Vert isPositive(inj(a,b)) + isNegative(inj(a,b)) \Vert$

for $a:\mathbf{2}$, $b:\mathbb{N}$.

###### Definition

The dependent type is less than, denoted as $inj(a,b) \lt inj(c,d)$, is defined as

$inj(a,b) \lt inj(c,d) \coloneqq isPositive(inj(c,d) - inj(a,b))$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$.

###### Definition

The dependent type is greater than, denoted as $inj(a,b) \gt inj(c,d)$, is defined as

$inj(a,b) \gt inj(c,d) \coloneqq isNegative(inj(c,d) - inj(a,b))$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$.

###### Definition

The dependent type is apart from, denoted as $inj(a,b) # inj(c,d)$, is defined as

$inj(a,b) # inj(c,d) \coloneqq isNonZero(inj(c,d) - inj(a,b))$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$.

###### Definition

The dependent type is less than or equal to, denoted as $inj(a,b) \leq inj(c,d)$, is defined as

$inj(a,b) \leq inj(c,d) \coloneqq isNonNegaitve(inj(c,d) - inj(a,b))$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$.

###### Definition

The dependent type is greater than or equal to, denoted as $inj(a,b) \geq inj(c,d)$, is defined as

$inj(a,b) \geq inj(c,d) \coloneqq isNonPositive(inj(c,d) - inj(a,b))$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$.

### Pseudolattice structure on the integers

###### Definition

The ramp function $ramp:\mathbb{Z} \to \mathbb{Z}$ is inductively defined as

$ramp(inj(0, a)) \coloneqq inj(0, a)$
$ramp(inj(1, a)) \coloneqq 0$

for $a:\mathbb{N}$.

###### Definition

The minimum $min:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is defined as

$min(inj(a,b),inj(c,d)) \coloneqq inj(a,b) - ramp(inj(a,b) - inj(c,d))$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$.

###### Definition

The maximum $max:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is defined as

$max(inj(a,b),inj(c,d)) \coloneqq inj(a,b) + ramp(inj(c,d) - inj(a,b))$

for $a:\mathbf{2}$, $b:\mathbb{N}$, $c:\mathbf{2}$, $d:\mathbb{N}$.

###### Definition

The absolute value $\vert(-)\vert:\mathbb{Z} \to \mathbb{Z}$ is defined as

$\vert inj(a,b) \vert \coloneqq max(inj(a,b), -inj(a,b))$

for $a:\mathbf{2}$, $b:\mathbb{N}$.

### Division and remainder

###### Definition

Integer division $(-)\div(-):\mathbb{Z} \times \mathbb{Z}_{\neq 0} \to \mathbb{Z}$ is defined as

$inj(0,a) \div inj(0,b) \coloneqq inj(0,a \div b)$
$inj(0,a) \div inj(1,b) \coloneqq inj(1,a \div b)$
$inj(1,a) \div inj(0,b) \coloneqq inj(1,a \div b)$
$inj(1,a) \div inj(1,b) \coloneqq inj(0,a \div b)$

for $a:\mathbb{N}$, $b:\mathbb{N}_{\neq 0}$.

## Properties

The integers are sequentially Cauchy complete with respect to the distance function defined as $d(x, y) \coloneqq \vert x - y \vert$. This is because any Cauchy sequence in the integers converges to an integer $n$ and only has a finite number of terms that are not equal to $n$. This is because the integers are not dense.