nLab list

Contents

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 =
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

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Deduction and Induction

Rules for types of lists

See also
References

Idea

In mathematics and specifically in formal logic, by a “list” one means the fairly evident formalization of the colloquial notion of a “list of elements”:

If a set $E$ of admissible elements is given — also called an “alphabet”, in this context — then a list of $E$ -elements is a tuple $(e_1, e_2, \cdots, e_\ell)$ of elements $e_i \,\in\, E$ , of any length $\ell \,\in\, \mathbb{N}$ — also called a word in the given alphabet.

Typically one will write “ $List(E)$ ” or similar for the set of all lists with entries in $E$ .

For instance, for $E$ the (underlying set of) a field, the component-expressions of vectors in the vector space $k^n$ may be thought of as lists of length $n$ with elements in $k$ . However, doing so means to disregard the vector space-structure on the collection of all such lists.

Instead, the notion of list is a concept with an attitude: While lists are just tuples of any length, calling them lists indicates that one intends to consider the operation of concatenation of lists to larger lists, in particular the operations of appending an element to a list.

It is evident that the operation of concatenation makes $List(E)$ a monoid (the neutral element under concatenation is the empty list, i.e. the unique list of length zero). In fact, a moment of reflection shows that, as such, $\big(List(E), conc\big)$ is the free monoid on $E$ .

Therefore, in much of the mathematical literature, lists are understood as free monoids.

Specifically in computer science one commonly deals with the corresponding notion of the data type of lists (with entries in a prescribed data type), which serve as a basic and common kind of data structure?. In programming languages supporting something like a calculus of constructions, this data type, essentially with the free monoid-structure as above, may be defined as an inductive type in an evident way (made explicit below).

On the other hand, in dependent type theories which have a homotopy type theory-interpretation — in that they do not verify uniqueness of identity proofs (or “axiom K”) — one may want to distinguish between lists and free monoids:

Because, in such theories it may happen that the given alphabet type $E$ is not actually a set but a higher homotopy type, in that it is not 0-truncated. In this case it still makes good sense to speak of lists of elements (terms) of $E$ , only that now the corresponding type $List(E)$ is itself not 0-truncated. But since the term “monoid” carries with it a connotation of being 0-truncated, one may no longer want to call this the free monoid on $E$ . It is, of course, still a free monoid in the proper higher algebraic sense (cf. monoid in a monoidal $\infty$ -category).

Definition

The type of lists on a type $A$ is defined as the dependent sum type

\mathrm{List}(A) \coloneqq \sum_{n:\mathbb{N}} \mathrm{Fin}(n) \to A

where $\mathrm{Fin}(n)$ is the finite set with $n$ elements. This shows that lists are just tuples.

Rules for types of lists

Assuming that identification types, function types and dependent product types exist in the type theory, the type of lists on a type $A$ of generators is the inductive type $\mathrm{List}(A)$ generated by an element $\mathrm{nil}:\mathrm{List}(A)$ and a function $\mathrm{cons}:A \to \mathrm{List}(A) \to \mathrm{List}(A)$ :

Formation rules for the type of lists:

\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{List}(A) \; \mathrm{type}}

Introduction rules for the type of lists:

\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{nil}:\mathrm{List}(A)} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{cons}:A \to (\mathrm{List}(A) \to \mathrm{List}(A))}

Elimination rules for the type of lists:

\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\ \Gamma \vdash g:\mathrm{List}(A)) \end{array} }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g):C(g)}

Computation rules for the type of lists:

Judgmental computation rules

\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \end{array} }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{nil}) \equiv c_\mathrm{nil}:C(\mathrm{nil})}

\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\ \Gamma \vdash b:A \quad \Gamma \vdash g:\mathrm{List}(A)) \end{array} }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{cons}(b)(g)) \equiv c_\mathrm{cons}(b)(g)(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g)):C(\mathrm{cons}(b)(g))}

Typal computation rules

\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \end{array} }{\Gamma \vdash \beta_{\mathrm{List}(A)}^\mathrm{nil}(c_\mathrm{nil}, c_\mathrm{cons}):\mathrm{Id}_{C(\mathrm{nil})}(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{nil}), c_\mathrm{nil})}

\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\ \Gamma \vdash b:A \quad \Gamma \vdash g:\mathrm{List}(A)) \end{array} }{\Gamma \vdash \beta_{\mathrm{List}(A)}^\mathrm{cons}(c_\mathrm{nil}, c_\mathrm{cons}, b, g):\mathrm{Id}_{C(\mathrm{cons}(b)(g))}(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{cons}(b)(g)), c_\mathrm{cons}(b)(g)(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g)))}

Uniqueness rules for the type of lists:

Judgmental uniqueness rules

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathrm{List}(A)} C(x) \quad \Gamma \vdash g:\mathrm{List}(A)}{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c(\mathrm{nil}), \lambda a:A.\lambda x:\mathrm{List}(A).c(\mathrm{cons}(a)(x)), g) \equiv c(g):C(g)}

Typal uniqueness rules

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathrm{List}(A)} C(x) \quad \Gamma \vdash g:\mathrm{List}(A)}{\Gamma \vdash \eta_{\mathrm{List}(A)}(c, n):\mathrm{Id}_{C(g)}(\mathrm{ind}_{\mathrm{List}(A)}^C(c(\mathrm{nil}), \lambda a:A.\lambda x:\mathrm{List}(A).c(\mathrm{cons}(a)(x)), g), c(g))}

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

\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x))}{\Gamma \vdash \mathrm{up}_\mathbb{Z}^C(c_\mathrm{nil}, c_\mathrm{cons}):\exists!c:\prod_{x:\mathrm{List}(A))} C(x).\mathrm{Id}_{C(\mathrm{nil})}(c(\mathrm{nil}), c_\mathrm{nil}) \times \prod_{a:A} \prod_{x:\mathrm{List}(A)} \mathrm{Id}_{C(\mathrm{cons}(a)(x))}(c(\mathrm{cons}(a)(x)), c_\mathrm{cons}(a)(c(x)))}

References

Types of lists are defined in section 5.1 in:

Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Project, Institute for Advanced Study, 2013. (web, pdf)

Last revised on January 25, 2023 at 04:24:46. See the history of this page for a list of all contributions to it.

nLab list

Context

Type theory

Deduction and Induction

Foundations

The basis of it all

Set theory

Foundational axioms

Removing axioms

Contents

Idea

Definition

Rules for types of lists

See also

References