nLab
axiom UIP

Contents

Context

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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
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
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) 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

semantics

Contents

Idea

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 propositionally extensional 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.

Statement

Definition

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)}

References

Discussion in Coq is in

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

Last revised on August 9, 2018 at 17:39:26. See the history of this page for a list of all contributions to it.