nLab proof theoretic strength of univalent type theory plus HITs



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


Constructivism, Realizability, Computability



The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



This is a stub collecting information on the proof theoretic strength of univalent type theory with higher inductive types.

Setzer computes the strength of MLTT+W. Voevodsky reduces Coq’s inductive types to W-types. Coquand et al reduces univalence and some HITs to an unspecified constructive framework, which can be taken to be CZF+REA with a sequence of universes. It follows from this work that univalent type theory and these HITs has the same proof theoretic strength as MLTT with inductive definitions and a sequence of universes.

There is an MO-question on the proof theoretic strength of pCIC. Avigad provide a general overview of the proof theory of predicative constructive systems.

The strength of type theories

This section collects some references that establish the strength various type theories, roughly in increasing order of strength.

Let ML n\mathrm{ML}_n denote MLTT without inductive types and nn universes closed under Π\Pi and Σ\Sigma. This has the strength of the first-order system of nn iterated fixed points (whether with intuitionistic or classical logic), and thus by Feferman 1982 has proof-theoretic ordinal ξ n\xi_n, where ξ 0=ε 0\xi_0=\varepsilon_0 and ξ n+1=φ(ξ n,0)\xi_{n+1} = \varphi(\xi_n,0).


  • Jeremy Avigad, Proof Theory, 2014 PDF

  • Thierry Coquand et al, Variation on Cubical sets, 2014 PDF.

  • Solomon Feferman, Iterated inductive fixed-point theories: application to Hancock’s conjecture, Stud. Logic Foundations Math., 109, 1982.

  • Anton Setzer, Proof theoretical strength of Martin-Lof Type Theory with W-type and one universe PDF.

  • Anton Setzer, Proof theory of Martin-Löf type theory. An overview PDF

  • Vladimir Voevodsky, Notes on type systems 2011 PDF.

Last revised on August 5, 2023 at 14:29:05. See the history of this page for a list of all contributions to it.