nLab intensional type theory

Redirected from "intensional type theories".

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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Decidability

Examples
Related concepts
References

Idea

By intensional type theory one means the flavor of type theory in which identity types are not forced to be judgmental equalities by equality reflection.

The original definition of identity types in Martin-Löf dependent type theory [Martin-Löf (1975, §1.7 and p. 94)] is by default intensional: It takes an additional axiom or rule — the identity reflection rule, cf. Streicher (1993, p. 4, 13 ); Hofmann (1995, p. 16) — to turn intensional into extensional type theory (in which identity types are then turned into judgmental equality).

In particular,

homotopy type theory is intensional: the categorical semantics of its identity types are path space objects.
Any theory with UIP or axiom K but no equality reflection is an intensional set-level type theory

Properties

Decidability

Only the intensional but not the extensional Martin-Löf type theory has decidable type checking. (Martin-Löf 75, Hofmann 95).

Examples

calculus of inductive constructions

Related concepts

References

Per Martin-Löf, An intuitionistic theory of types: predicative part, in: H. E. Rose, J. C. Shepherdson (eds.), Logic Colloquium ‘73, Proceedings of the Logic Colloquium, Studies in Logic and the Foundations of Mathematics 80 Pages 73-118, Elsevier 1975 (doi:10.1016/S0049-237X(08)71945-1, CiteSeer)
Thomas Streicher, Investigations into Intensional Type Theory, Habilitation Thesis, Darmstadt (1993) [pdf, pdf]
Martin Hofmann, Extensional concepts in intensional type theory, Ph.D. thesis, University of

Edinburgh, (1995) (ECS-LFCS-95-327, pdf)

Intensional type theory is discussed in chapter 4 of:

Carlo Angiuli, Daniel Gratzer, Principles of Dependent Type Theory (pdf)

Last revised on May 17, 2025 at 13:47:00. See the history of this page for a list of all contributions to it.