nLab type refinement



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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject 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




Types enable the restriction of the domain and codomain of computer programs, but they may not always be precise enough. Refining the type system of a language involves introducing subtypes while retaining the syntax of its terms. For instance, predicate subtyping can be used to limit head to non-empty lists (list{ l : len(l) > 0}) and div to non-zero arguments (int{ n : n <> 0 }).

More precisely, in Greenberg 2015, Frank Pfenning describes type refinement as a long-term research program with the following goals:

  • capture more precise properties of programs,

  • retain the good theoretical and practical properties of the simpler disciplines,

  • retain usability, modularity, elegance, etc.

Works in this program include:

  • datasort refinements (types are named refinement types) of Freeman, Davies, and Pfenning, which refine the set of available constructors for a type;

  • predicate subtyping (types are named refined types or predicate subtypes), where predicates are taken from a tractable domain (see for example index refinements of Xi and Pfenning);

but don’t include general subset types?, as type checking becomes undecidable.

Predicate subtyping is found in languages like Liquid Haskell? and F*?.


On datasort refinements:

  • Tim Freeman, Frank Pfenning, Refinement types for ML, Proceedings of the ACM Conference on Programming Language Design and Implementation, 1991, pp. 268–277, (pdf).
  • Frank Pfenning, Church and Curry: Combining Intrinsic and Extrinsic Typing, (pdf).

On predicate subtyping:

  • Ranjit Jhala, Niki Vazou, Refinement Types: A Tutorial, (arXiv:2010.07763).
  • Yitzhak Mandelbaum, David Walker, Robert Harper, An Effective Theory of Type Refinements, (pdf).
  • Susumu Hayashi, Logic of refinement types, Proceedings of the Workshop on Types for Proofs and Programs, 1993, pp. 157-172.

On index refinements in particular:

  • Hongwei Xi, Frank Pfenning, Dependent types in practical programming, in A. Aiken, editor, Conference Record of the 26th Symposium on Principles of Programming Languages (POPL’99),pages 214–227. ACM Press, January 1999.

On refinement systems as functors:

Last revised on December 19, 2023 at 19:17:52. See the history of this page for a list of all contributions to it.