nLab sum type (changes)

Showing changes from revision #12 to #13: Added | ~~Removed~~ | ~~Chan~~ged

Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

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

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

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

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

Edit this sidebar

Idea
Definition

Properties

Descent and large elimination

Related concepts
References

Idea

In type theory a sum type of two types $A$ and $B$ (really: coproduct) is the type whose terms are either terms $a\colon A$ or terms $b\colon B$ .

In a model of the type theory in categorical semantics this is a distributive coproduct. In set theory, it is a disjoint union.

Definition

Like all type constructors in type theory, to characterize sum types we must specify how to build them, how to construct elements of them, how to use such elements, and the computation rules.

The way to build sum types is easy:

\frac{A\colon Type \qquad B\colon Type}{A+B \colon Type}

As a positive type

Sum types are most naturally presented as positive types, so that the constructor rules are primary. These say that we can obtain an element of $A+B$ from an element of $A$ , or from an element of $B$ .

\frac{a\colon A}{inl(a)\colon A+B} \qquad \frac{b\colon B}{inr(b)\colon A+B}

The eliminator is derived from these: it says that in order to use an element of $A+B$ , it suffices to specify what should be done for the two ways in which that element could have been constructed.

\frac{p\colon A+B \qquad x\colon A\vdash c_A\colon C \qquad y\colon B \vdash c_B\colon C}{match(p, x.c_A, y.c_B) \colon C}

The terms $c_A$ and $c_B$ can have free variables $x$ and $y$ respectively, but those variables become bound in the $match$ expression. In dependent type theory, we must generalize the eliminator to allow $C$ to depend on $A+B$ .

The beta reduction rules for a constructor followed by an eliminator:

\begin{aligned} match(inl(a), x.c_A, y.c_B) &\to_\beta c_A[a/x]\\ match(inr(b), x.c_A, y.c_B) &\to_\beta c_B[b/y] \end{aligned}

The eta reduction rule for the opposite composite says that for any term $c\colon C$ in the context of $p\colon A+B$ ,

match(p, x.c[inl(x)/z], y.c[inr(y)/z]) \to_\eta c[p/z].

This says that if we unpack a term of type $A+B$ , but only use the resulting term of type $A$ or $B$ by way of packing them back into $A+B$ , then we might as well not have unpacked them to begin with. Note that choosing $C\coloneqq A+B$ and $c \coloneqq z$ , we obtain a simpler form of $\eta$ -conversion:

match(p, x.inl(x), y.inr(y)) \to_\eta p.

The positive presentation of sum types can be regarded as a particular sort of inductive type. In Coq syntax:

Inductive sum (A B:Type) : Type :=
| inl : A -> sum A B
| inr : B -> sum A B.

Coq implements the beta reduction rule, but not the eta (although eta equivalence is provable for the inductively defined identity types, using the dependent eliminator mentioned above).

As a negative type

It is possible to present sum types as negative types as well, but only if we allow sequents with multiple conclusions. This is common in sequent calculus presentations of classical logic, but not as common in type theory and almost unheard of in dependent type theory.

The two definitions are provably equivalent, but only using the contraction rule and the weakening rule. Thus, in linear logic they become distinct; the positive sum type is “plus” $A\oplus B$ and the negative one is “par” $A \parr B$ .

With typal computation and uniqueness rules

Assuming that identification types, function types and dependent product types exist in the type theory, the sum type of types $A$ and $B$ is the inductive type generated by a function from $A$ to $A + B$ and a function from $B$ to $A + B$ :

Formation rules for sum types:

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

Introduction rules for sum types:

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{in}_A:A \to A + B} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{in}_B:B \to A + B}

Elimination rules for sum types:

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_{\mathrm{in}_A}:\prod_{x:A} C(\mathrm{in}_A(x)) \quad \Gamma \vdash c_{\mathrm{in}_B}:\prod_{y:B} C(\mathrm{in}_B(y)) \quad \Gamma \vdash z:A + B}{\Gamma \vdash \mathrm{ind}_{A + B}^C(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, z):C(y)}

Computation rules for sum types:

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_{\mathrm{in}_A}:\prod_{x:A} C(\mathrm{in}_A(x)) \quad \Gamma \vdash c_{\mathrm{in}_B}:\prod_{y:B} C(\mathrm{in}_B(y)) \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_{A + B}^{\mathrm{in}_A}(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, a):\mathrm{Id}_{C(\mathrm{in}_A(a))}(\mathrm{ind}_{A + B}^C(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, \mathrm{in}_A(a)), c(a))}

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_{\mathrm{in}_A}:\prod_{x:A} C(\mathrm{in}_A(x)) \quad \Gamma \vdash c_{\mathrm{in}_B}:\prod_{y:B} C(\mathrm{in}_B(y)) \quad \Gamma \vdash b:B}{\Gamma \vdash \beta_{A + B}^{\mathrm{in}_B}(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, b):\mathrm{Id}_{C(\mathrm{in}_B(b))}(\mathrm{ind}_{A + B}^C(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, \mathrm{in}_B(b)), c(b))}

Uniqueness rules for sum types:

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:A + B} C(x) \quad \Gamma \vdash z:A + B}{\Gamma \vdash \eta_{A + B}(c, z):\mathrm{Id}_{C(z)}(\mathrm{ind}_{A + B}^C(\lambda x:A.c(\mathrm{in}_A(x)), \lambda y:B.c(\mathrm{in}_B(y)), z), c(z))}

The elimination, computation, and uniqueness rules for the sum type of $A$ and $B$ state that the sum type of $A$ and $B$ satisfy the dependent universal property of the sum type of $A$ and $B$ . 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 sum type of of $A$ and $B$ could be simplified to the following rule:

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_{\mathrm{in}_A}:\prod_{x:A} C(\mathrm{in}_A(x)) \quad \Gamma \vdash c_{\mathrm{in}_B}:\prod_{y:B} C(\mathrm{in}_B(y))}{\Gamma \vdash \mathrm{up}_{A + B}^C(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}):\exists!c:\prod_{x:A + B} C(x).\left(\prod_{a:A} \mathrm{Id}_{C(\mathrm{in}_A(a))}(c(\mathrm{in}_A(a)), c_{\mathrm{in}_A}(a))\right) \times \left(\prod_{b:B} \mathrm{Id}_{C(\mathrm{in}_B(b))}(c(\mathrm{in}_B(b)), c_{\mathrm{in}_B}(b))\right)}

In terms of dependent sum types and booleans

The sum type can be defined in terms of the boolean domain and the dependent sum type. Given types $A$ and $B$ , the sum type $A + B$ is defined as

A + B \coloneqq \sum_{x:\mathrm{Bool}} ((x =_\mathrm{Bool} 1) \to A) \times ((x =_\mathrm{Bool} 0) \to B)

Properties

Descent and large elimination

The descent for the sum type $A + B$ states that given any type families $x:A \vdash C(x)$ and $y:B \vdash D(y)$ one can construct a type family $z:A + B \vdash \mathrm{descFam}_{A + B}^{C, D}(z)$ with families of equivalences of types

x:A \vdash \mathrm{descEquiv}_{A + B}^C:\mathrm{descFam}_{A + B}^{C, D}(\mathrm{inl}(x)) \simeq C(x)

y:B \vdash \mathrm{descEquiv}_{A + B}^D:\mathrm{descFam}_{A + B}^{C, D}(\mathrm{inr}(y)) \simeq D(y)

Large elimination for sum types strengthens the equivalences of types in descent to judgmental equality of types

x:A \vdash \mathrm{descFam}_{A + B}^{C, D}(\mathrm{inl}(x)) \equiv C(x) \; \mathrm{type}

y:B \vdash \mathrm{descFam}_{A + B}^{C, D}(\mathrm{inr}(y)) \equiv D(y) \; \mathrm{type}

Related concepts

References

A textbook account in the context of programming languages is in section 12 of

Robert Harper, Practical Foundations for Programming Languages

For sum types in homotopy type theory, see:

Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Project, Institute for Advanced Study, 2013. (web, pdf)
Egbert Rijke, Introduction to Homotopy Type Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press (arXiv:2212.11082)

Last revised on February 27, 2024 at 21:37:37. See the history of this page for a list of all contributions to it.

nLab sum type (changes)

Context

Type theory

Contents

Idea

Definition

As a positive type

As a negative type

With typal computation and uniqueness rules

In terms of dependent sum types and booleans

Properties

Descent and large elimination

Related concepts

References