nLab alpha-equivalence


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


Equality and Equivalence



In logic and type theory, α\alpha-equivalence is the principle that two syntactic expressions (types, terms, propositions, contexts, whatever) are equivalent for all purposes if their only difference is the renaming of bound variables.

Depending on the technicalities of how variables are managed, α\alpha-equivalence may be a necessary axiom, a provable theorem, or entirely trivial. In any case, it is often (usually? always?) seen as a technicality devoid of conceptual interest. However, it can be a technically nontrivial task to implement α\alpha-conversion (which is necessary to avoid capture of free variables upon substitution) when programming logic or type theory into a computer.


We work in a dependent type theory which acts as the metatheory for our object type theory.

The type 𝔸\mathbb{A} is a countable set (and thus a type with decidable equality) whose terms are called names or atoms. The type Exp\mathrm{Exp} of syntactic expressions is an h-set which is inductively defined by binary functions P:Exp×ExpExpP:\mathrm{Exp} \times \mathrm{Exp} \to \mathrm{Exp} and B([()]()):𝔸×ExpExpB([(-)](-)):\mathbb{A} \times \mathrm{Exp} \to \mathrm{Exp}. There are dependent types E:ExpocctypeE:\mathrm{Exp} \vdash \mathrm{occ} \; \mathrm{type} representing the occuring atoms, E:ExpfreetypeE:\mathrm{Exp} \vdash \mathrm{free} \; \mathrm{type} representing the free atoms, and E:ExpbndtypeE:\mathrm{Exp} \vdash \mathrm{bnd} \; \mathrm{type} representing the bound atoms, such that for each expression E:ExpE:\mathrm{Exp}, there are embeddings

i occ(E):occ(E)𝔸i_\mathrm{occ}(E):\mathrm{occ}(E) \hookrightarrow \mathbb{A}
i free(E):free(E)𝔸i_\mathrm{free}(E):\mathrm{free}(E) \hookrightarrow \mathbb{A}
i bnd(E):bnd(E)𝔸i_\mathrm{bnd}(E):\mathrm{bnd}(E) \hookrightarrow \mathbb{A}

which may be defined by recursion on Exp\mathrm{Exp}.

We define the following dependent types a:𝔸,E:ExpaEtypea:\mathbb{A}, E:\mathrm{Exp} \vdash a \lhd E \; \mathrm{type} as

aE x:occ(E)i occ(E)(x)= 𝔸aa \lhd E \coloneqq \sum_{x:\mathrm{occ}(E)} i_\mathrm{occ}(E)(x) =_\mathbb{A} a

representing that aa occurs in EE at least once,

a freeE x:free(E)i free(E)(x)= 𝔸aa \lhd_\mathrm{free} E \coloneqq \sum_{x:\mathrm{free}(E)} i_\mathrm{free}(E)(x) =_\mathbb{A} a

representing that aa freely occurs in EE at least once, and

a#E(a freeE)a \# E \coloneqq (a \lhd_\mathrm{free} E) \to \emptyset

representing that aa is fresh (does not freely occurs in EE at least once)

See also


The distinction between α\alpha-equivalence and syntactic equality of expressions is briefly discussed in:

Last revised on November 11, 2022 at 04:16:31. See the history of this page for a list of all contributions to it.