Homotopy Type Theory integers > history (changes)

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Contents

< integer

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:2×inj : \mathbf{2} \times \mathbb{N} \rightarrow \mathbb{Z}.
  • An identity representing that positive and negative zero are equal: ν 0:inj(0,0)=inj(1,0)\nu_0: inj(0, 0) = inj(1, 0).

Definition 2

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

  • A term 0:0 : \mathbb{Z}.
  • A function s:s : \mathbb{Z} \to \mathbb{Z}.
  • A function p 1:p_1 : \mathbb{Z} \to \mathbb{Z}.
  • A function p 2:p_2 : \mathbb{Z} \to \mathbb{Z}.
  • A dependent product of identities representing that p 1p_1 is a section of ss:
    σ: a:p 1(s(a))=a\sigma: \prod_{a:\mathbb{Z}} p_1(s(a)) = a
  • A dependent product of identities representing that p 2p_2 is a retracion of ss:
    ρ: a:s(p 2(a))=a\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 0:0 : \mathbb{Z}.
  • A function s:s : \mathbb{Z} \to \mathbb{Z}.
  • A function p:p : \mathbb{Z} \to \mathbb{Z}.
  • A dependent product of identities representing that pp is a section of ss:
    σ: a:p(s(a))=a\sigma: \prod_{a:\mathbb{Z}} p(s(a)) = a
  • A dependent product of identities representing that pp is a retracion of ss:
    ρ: a:s(p(a))=a\rho: \prod_{a:\mathbb{Z}} s(p(a)) = a
  • A dependent product of identities representing the coherence condition:
    κ: a:ap s(σ(a))=ρ(a)\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 0:0 : \mathbb{Z}.
  • A function s:s : \mathbb{Z} \to \mathbb{Z}.
  • A function n:n : \mathbb{Z} \to \mathbb{Z}.
  • An identity representing that zero and negative zero are equal: ν:n(0)=0\nu: n(0) = 0.
  • A dependent product of identities representing that negation is an involution:
    ι: a:n(n(a))=a\iota: \prod_{a:\mathbb{Z}} n(n(a)) = a
  • A dependent product of identities representing the coherence condition for the above:
    κ ι: a:ap n(ι(a))=ι(a)\kappa_\iota: \prod_{a:\mathbb{Z}} ap_n(\iota(a)) = \iota(a)
  • A dependent product of identities representing that nsnn \circ s \circ n is a section of ss:
    σ: a:n(s(n(s(a))))=a\sigma: \prod_{a:\mathbb{Z}} n(s(n(s(a)))) = a
  • A dependent product of identities representing that nsnn \circ s \circ n is a retracion of ss:
    ρ: a:s(n(s(n(a))))=a\rho: \prod_{a:\mathbb{Z}} s(n(s(n(a)))) = a
  • A dependent product of identities representing the coherence condition:
    κ: a:ap s(σ(a))=ρ(a)\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:×inj : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{Z}.
  • A dependent product of functions between identities representing that equivalent differences are equal:
    equivdiff: a: b: c: d:(a+d=c+b)(inj(a,b)=inj(c,d))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
    τ 0: a: b:isProp(a=b)\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 0:0:\mathbb{Z} is defined as

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

Let ()+():×(-)+(-):\mathbb{N} \times \mathbb{N} \to \mathbb{N} be addition of the natural numbers, let ρ:×\rho:\mathbb{N} \times \mathbb{N} \to \mathbb{N} be the symmetric difference or metric of the natural numbers, and let <:×𝟚\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)inj(0,a+b)inj(0,a) + inj(0,b) \coloneqq inj(0,a+b)
inj(1,a)+inj(1,b)inj(1,a+b)inj(1,a) + inj(1,b) \coloneqq inj(1,a+b)
inj(0,a)+inj(1,b)inj(a<b,ρ(a,b))inj(0,a) + inj(1,b) \coloneqq inj(a \lt b,\rho(a,b))
inj(1,a)+inj(0,b)inj(b<a,ρ(a,b))inj(1,a) + inj(0,b) \coloneqq inj(b \lt a,\rho(a,b))

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

Definition

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

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

for a:2a:\mathbf{2}, b:b:\mathbb{N}, ¬:22\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)inj(a,b)+inj(¬c,d)inj(a, b) - inj(c, d) \coloneqq inj(a, b) + inj(\neg c, d)

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

Definition

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

1inj(0,1)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)inj(c,d)inj(ac,bd)inj(a,b) \cdot inj(c,d) \coloneqq inj(a \oplus c,b \cdot d)

for a:2a:\mathbf{2}, b:b:\mathbb{N}, c:2c:\mathbf{2}, d:d:\mathbb{N}, ()():×(-)\cdot(-):\mathbb{N} \times \mathbb{N} \to \mathbb{N}, ()():2×22(-)\oplus(-):\mathbf{2} \times \mathbf{2} \to \mathbf{2} (exclusive or binary operation in 2\mathbf{2}).

Definition

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

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

for a:2a:\mathbf{2}, b:b:\mathbb{N}, 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))isPositive(inj(a,b)), is defined as

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

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

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Pseudolattice structure on the integers

Definition

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

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

for a:a:\mathbb{N}.

Definition

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

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

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

Definition

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

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

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

Definition

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

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

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

Division and remainder

Definition

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

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

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

Properties

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

See also

References

Last revised on June 17, 2022 at 18:15:40. See the history of this page for a list of all contributions to it.