nLab uniqueness of identity proofs



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
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit 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 setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
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




In type theory what is called the UIP axiom, the axiom of uniqueness of identity proofs is an axiom that when added to intensional type theory turns it into a set-level type theory.

The axiom asserts that any two terms of the same identity type Id A(x,y)Id_A(x,y) are themselves propositionally equal (in the corresponding higher identity type).

This is contrary to the univalence axiom, which makes sense only in the absence of UIP.



The UIP axiom (for types in a universeTypeType”) posits that the type

A:Typex,y:Ap,q:Id A(x,y)Id Id A(x,y)(p,q) \underset{A \colon Type}{\prod} \underset{x,y \colon A}{\prod} \underset{p,q \colon Id_A(x,y)}{\prod} Id_{Id_A(x,y)}(p,q)

has a term. In other words, we add to our type theory the rule

uip:A:Typex,y:Ap,q:Id A(x,y)Id Id A(x,y)(p,q) \frac{}{ \vdash uip \colon \underset{A \colon Type}{\prod} \underset{x,y \colon A}{\prod} \underset{p,q \colon Id_A(x,y)}{\prod} Id_{Id_A(x,y)}(p,q)}

We can modify this by making the hypotheses of the axiom into premises of the rule, if we wish. In particular, this can be used to give a version of the rule that applies to all types not necessarily belonging to some fixed universe, using the judgmentAtypeA\;type” for “AA is a type” (as distinguished from “A:TypeA:Type” for “AA belongs to the universe type TypeType”).

ΓAtypeΓx:AΓy:AΓp:Id A(x,y)Γq:Id A(x,y)Γuip:Id Id A(x,y)(p,q) \frac{\Gamma\vdash A\; type \quad \Gamma\vdash x : A \quad \Gamma \vdash y:A \quad \Gamma \vdash p : Id_A(x,y) \quad \Gamma \vdash q:Id_A(x,y)}{ \Gamma\vdash uip : Id_{Id_A(x,y)}(p,q)}

Definitional UIP

There is also a definitional version of UIP, where any two terms of the same identity type or path type are definitionally equal. It is given by the following rule:

ΓAtypeΓx:AΓy:AΓp:Id A(x,y)Γq:Id A(x,y)Γpq:Id A(x,y) \frac{\Gamma\vdash A\; type \quad \Gamma\vdash x : A \quad \Gamma \vdash y:A \quad \Gamma \vdash p : Id_A(x,y) \quad \Gamma \vdash q:Id_A(x,y)}{\Gamma\vdash p \equiv q: Id_A(x,y)}

Definitional UIP is implied by boundary separation in cubical type theory.


Discussion in Coq is in

  • Pierre Corbineau, The K axiom in Coq (almost) for free (pdf)

For definitional UIP in XTT

Last revised on September 3, 2022 at 12:05:25. See the history of this page for a list of all contributions to it.