nLab quotient inductive type

Contents

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition, truth value(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
set of truth valuessubobject classifiertype of propositions
universeobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Induction

Contents

Idea

Quotient inductive types are higher inductive types with a 0-truncation constructor.

Examples

All quotient inductive types described below are given together with some pseudo-Coq code, which would implement that QIT if Coq supported QITs natively.

Integers

A definition of the set of integers as a quotient inductive type.

Inductive int : Type :=
| zero : int
| succ : int -> int
| pred : int -> int
| sec : forall (x : int) pred succ x == x
| ret : forall (x : int) succ pred x == x
| contr1 : forall (x y : int) (p q : x == y), p == q.

Another definition of the integers as a quotient inductive type:

Inductive int : Type :=
| zero : int
| succ : int -> int
| neg : int -> int
| axiom1 : neg zero  == zero
| axiom2 : forall (x : int) succ neg succ x == neg x
| contr1 : forall (x y : int) (p q : x == y), p == q.

Dyadic rational numbers

A minimal inductive definition of the dyadic rational numbers as a quotient inductive type:

Inductive dyadic : Type :=
| zero : dyadic
| succ : dyadic -> dyadic
| pred : dyadic -> dyadic
| act : nat -> dyadic -> dyadic
| sec : forall (x : int) pred succ x == x
| ret : forall (x : int) succ pred x == x
| initfunc : forall (x : dyadic) act 0 x == succ x
| funcsqrt : forall (n : nat) forall (x : dyadic) act (succ n) (act (succ n) x) == act n x
| contr1 : forall (x y : dyadic) (p q : x == y), p == q.

A definition of the dyadic rationals as a symmetric midpoint algebra

Inductive dyadic : Type :=
| zero : dyadic
| succ : dyadic -> dyadic
| neg : dyadic -> dyadic
| mid : dyadic -> dyadic -> dyadic
| inv : forall (x : dyadic) neg neg x == x
| idem : forall (x : dyaduc) mid x x == x
| comm : forall (x y : dyadic) mid x y == mid y x
| medial : forall (w x y z : dyadic) mid (mid w x) (mid y z) = mid (mid w y) (mid x z)
| neutr : forall (x : dyadic) mid (neg x) x == zero
| distneg : forall (x y : dyadic) mid (neg x) (neg y) == neg mid x y
| distsucc : forall forall (x y : dyadic) mid (succ x) (succ y) == succ mid x y
| twicesucc : forall forall (x y : dyadic) mid (succ succ x) y == succ mid x y
| contr1 : forall (x y : dyadic) (p q : x == y), p == q.

Classically extended natural numbers

The set of extended natural numbers as it behaves in classical mathematics as a quotient inductive type

Inductive extnat : Type :=
| zero : extnat
| succ : extnat -> extnat
| inf : extnat
| fixpoint : succ inf == inf
| contr1 : forall (x y : int) (p q : x == y), p == q.

Polynomial rings over integers

A definition of the polynomial ring over the integers with a set of indeterminants AA.

Inductive intpoly (A: Type)  :=
| indet : A -> (intpoly A)
| zero : intpoly A
| one : intpoly A
| neg : (intpoly A) -> (intpoly A)
| add : (intpoly A) -> (intpoly A) -> (intpoly A)
| mult : (intpoly A) -> (intpoly A) -> (intpoly A)
| alunital : forall (x : intpoly A) add zero x == x
| arunital : forall (x : intpoly A) add x zero == x
| aassoc : forall (x y z : intpoly A) add x (add y z) == add (add x y) z
| acomm : forall (x y : intpoly A) add x y = add y x
| alinv : forall (x : intpoly A) add (neg x) x == zero
| arinv : forall (x : intpoly A) add x (neg x) == zero
| ldist : forall (x y z : intpoly A) mult x (add y z) == add (mult x y) (mult x z)
| rdist : forall (x y z : intpoly A) mult (add x y) z == add (mult x z) (mult y z)
| mlunital : forall (x : intpoly A) mult one x == x
| mrunital : forall (x : intpoly A) mult x one == x
| massoc : forall (x y z : intpoly A) mult x (mult y z) == mult (mult x y) z
| mcomm : forall (x y : intpoly A) mult x y = mult y x
| contr1 : forall (x y : intpoly A) (p q : x == y), p == q.

Quotients of sets

The quotient of an hProp-valued or (-1)-truncated equivalence relation, yielding an hSet or 0-truncated type:

Inductive quotient (A : Type) (R : A -> A -> hProp) : Type :=
| proj : A -> quotient A R
| relate : forall (x y : A), R x y -> proj x == proj y
| contr1 : forall (x y : quotient A R) (p q : x == y), p == q.

Sierpinski space

A definition of Sierpinski space:

Inductive sierpinski : Type :=
| top : sierpinski
| meet : sierpinski -> sierpinski -> sierpinski
| meet-left-unital : forall (a : sierpinski) meet top a = a
| meet-right-unital : forall (a : sierpinski) meet a top = a
| meet-associative : forall (a b c : sierpinski) meet meet a b c = meet a meet b c
| meet-commutative : forall (a b : sierpinski) meet a b = meet b a
| meet-idempotent : forall (a : sierpinski) meet a a = a
| bottom : sierpinski
| join : sierpinski -> sierpinski -> sierpinski
| join-left-unital : forall (a : sierpinski) join bottom a = a
| join-right-unital : forall (a : sierpinski) join a bottom = a
| join-associative : forall (a b c : sierpinski) join join a b c = join a join b c
| join-commutative : forall (a b : sierpinski) join a b = join b a
| join-idempotent : forall (a : sierpinski) join a a = a
| meet-join-absorption : forall (a b : sierpinski) meet a join a b = a
| join-meet-absorption : forall (a b : sierpinski) join a meet a b = a
| sequence-join : (nat -> sierpinski) -> sierpinski
| sequence-bounded-above : forall (n : nat) (s : nat -> sierpinski) meet s n sequence-join s = s n
| sequence-least-upper-bound : forall (a : sierpinski) (s : nat -> sierpinski) (forall (n : nat) meet s n a = s n) -> (meet sequence-join s x = sequence-join s)
| element-sequence-meet : (sierpinski) -> (nat -> sierpinski) -> (nat -> sierpinski)
| p : forall (a : sierpinski) (s : nat -> sierpinski) element-sequence-meet a s n = meet a s n
| distributive : forall (a : sierpinski) (s : nat -> sierpinski) meet a sequence-join s = sequence-join element-sequence-meet a s 
| contr1 : forall (x y : int) (p q : x == y), p == q.

Cumulative hierarchy

According to Homotopy Type Theory – Univalent Foundations of Mathematics the cumulative hierarchy of material sets could be constructed from a type universe as a quotient inductive type.

Cauchy complete real numbers

According to Homotopy Type Theory – Univalent Foundations of Mathematics the Cauchy complete real numbers could be constructed from a type universe as a quotient inductive-inductive type.

Permutable trees

Altenkirch et al. gives the following definition of a permutable tree:

Inductive tree (A : Type) :=
| leaf : tree A
| node : (A -> tree A) -> tree A
| mix : (forall (e : A -> A) isequiv(e)) -> (forall (e : A -> A) forall (f: A -> tree A) node f = node(f \circ e)
| contr1 : forall (x y : tree A) (p q : x == y), p == q.

Free posets

A definition of a free poset as a quotient inductive-inductive type:

Inductive poset (A: Type) :=
| inj: A -> poset A
| antisym: (forall x y : poset A) (x \leq y) -> (y \leq x) -> (x == y)
| setcontr: forall (x y: poset A) (p q : x == y), p == q.

Inductive \leq {poset A}: poset A -> poset A -> Type :=
| refl: (forall x: poset A) x \leq x
| trans: (forall x y z: poset A) (x \leq y) -> (y \leq z) -> (x \leq z)
| pordercontr: forall (x y : poset A) (p q : x \leq y), p == q.

Free pointed omega-complete partial orders

It is possible to define the free pointed omega-complete partial order on a 0-truncated type AA as a quotient inductive type. They are usually abbreviated as free pointed ω\omega-cpos (see Altenkirch, Danielsson, and Kraus 2017, §3.2)

Inductive omegacpo (A: Type) :=
| inj: A -> omegacpo A
| bottom: omegacpo A
| denumjoin: (exists a: nat -> omegacpo A) (forall n: nat) a n \leq a succ n
| antisym: (forall x y : omegacpo A) (x \leq y) -> (y \leq x) -> (x == y)
| setcontr: forall (x y: omegacpo A) (p q : x == y), p == q.

Inductive \leq {omegacpo A}: omegacpo A -> omegacpo A -> Type :=
| refl: (forall x: omegacpo A) x \leq x
| trans: (forall x y z: omegacpo A) (x \leq y) -> (y \leq z) -> (x \leq z)
| init: (forall x : omegacpo A) bottom \leq x
| upperbound: (forall a: nat -> omegacpo A) ((forall n: nat) a n) \leq denumjoin a p
| leastupper: (forall x : omegacpo A) ((forall a: nat -> omegacpo A) ((forall n: nat) a n) \leq x) -> (forall a: nat -> omegacpo A) denum a p \leq x)
| pordercontr: forall (x y : poset A) (p q : x \leq y), p == q.

Examples include the Sierpinski space 1 1_\bot.

Internal type theory

From Altenkirch and Kaposi 2017

References

Last revised on September 26, 2022 at 10:11:00. See the history of this page for a list of all contributions to it.