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  1. Page created, but author did not leave any comments.


    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2022

    i can’t parse the displayed equation in the Definition section:

    What is “a= B(b)ba =_{B(b)} b”?, given that a,b:Aa,b \colon A

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMay 21st 2022

    Fixed parentheses

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMay 21st 2022

    Moved some of the discussion to idea section, clarified some things

    diff, v3, current

  2. started section on path types in cubical type theory


    diff, v5, current

  3. adding “dependent path type” and “heterogeneous path type” as alternate names for dependent identity type


    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeSep 4th 2022

    rolled back changes, see the discussion on identity type.

  4. added redirects for dependent identification and dependent identifications


    diff, v7, current

  5. added redirects


    diff, v8, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2022
    • (edited Oct 19th 2022)

    I have fixed the title of the two references (there was a spurious “a” in both cases, also the italicization in the second item was missing).

    I find the page name “dependent identity type” misleading, since also the standard ML identity type is, of course, dependent.

    The alternative “heterogenous identity type” seems better. One could also try “relative identity type”.

    diff, v9, current

  6. renaming this page as proposed by Urs Schreiber on nForum discussion page


    diff, v17, current

  7. I have rewritten the Definitions section of this article. The old Definitions section gives inference rules for the heterogeneous identity types, but those rules require that function types and dependent function types be defined beforehand, in addition to identity types. The new inference rules here are simpler in that they do not require any other type to be defined beforehand, only identity types.

    In addition, the dependent application to identifications for dependent functions are defined the usual way, by induction on reflexivity of the index type’s identity type

    Ionel Niculescu

    diff, v26, current

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeSep 14th 2023

    Prompted by Madeleine Birchfield’s mention of this page on the HoTT Zulip, I cleaned up a lot of nonsense.

    • Dependent type theory is not sequent calculus, even though its judgments-in-context use a similar notation to sequents.
    • There are only two possible computation rules: judgmental/definitional and propositional/typal. The page typal equality seems very confused and should also be dealt with.
    • The eta rule was wrong: it’s not about uniqueness of single elements of CC but about functions/open terms in CC depending on an identity. However, since eta-rules for identity types collapse the type theory to become extensional, I don’t think it is worthwhile even writing them out here, so I just deleted those.
    • Removed all references to “non-dependent heterogeneous identity types” which I think are useless.

    diff, v27, current

    • CommentRowNumber14.
    • CommentAuthorJames
    • CommentTimeSep 15th 2023

    started section on the role of heterogeneous identity types in the inference rules for higher inductive types

    diff, v28, current

  8. Theo Winterhalter talks about heterogeneous equality in his talk at DutchCATS a few months ago:

    • Théo Winterhalter, A conservative and constructive translation from extensional type theory to weak type theory, Strength of Weak Type Theory, DutchCATS, 11 May 2023. (slides)


    diff, v29, current

    • CommentRowNumber16.
    • CommentAuthorMadeleine Birchfield
    • CommentTimeSep 16th 2023
    • (edited Sep 16th 2023)

    Winterhalter’s heterogeneous identity types aren’t the same types as the heterogeneous identity types described in this article.

    Here, the heterogeneous identity types c= B pdc =_{B}^{p} d take in as input a type AA, a type family B:ATypeB:A \to \mathrm{Type}, elements a:Aa:A, b:Ab:A, c:B(a)c:B(a), d:B(b)d:B(b), and an identification p:a= Abp:a =_A b. In addition, there is only one constructor, which is the hrefl B(a,c):c= B refl A(a)chrefl_B(a, c):c =_{B}^{\mathrm{refl}_A(a)} c.

    Winterhalter’s heterogeneous identity types aA= Bba \;{_A}=_B b take in as input two types AA and BB and two elements a:Aa:A and b:Bb:B, and in addition to the constructor hrefl A(a):aA= Aahrefl_A(a):a \;{_A}=_A a it also seems to have a transport constructor heqtr(p,a):aA= Btr(p,a)heqtr(p, a):a \;{_A}=_B tr(p, a) for p:A= TypeBp:A =_{Type} B and a:Aa:A.

    • CommentRowNumber17.
    • CommentAuthorMadeleine Birchfield
    • CommentTimeSep 16th 2023
    • (edited Sep 16th 2023)

    I asked on the HoTT discord and apparently the version of heterogeneous identity types which appeared on Théo Winterhalter slides predates the ones given in the article, first being defined pre-HoTT by Conor McBride in his phd thesis titled “Dependently Typed Programs and their Proofs” - where he calls those types “John Major equality”. The heterogeneous identity types which appear in the article seem to be a post-HoTT thing.

  9. Added rules for the heterogeneous identity types which compare elements between two types AA and BB, rather than elements between a family of types.


    diff, v30, current

  10. Mentioned the use of identifications p:A= UBp:A =_U B instead of equivalences ABA \simeq B in the heterogeneous identity types between AA and BB and how they are equivalent under the univalence axiom for UU.


    diff, v30, current

  11. Moved Theo Winterhalter reference to newly created article John Major equality


    diff, v31, current

  12. fixing formation rule hypothesis ΓB(x)type\Gamma \vdash B(x) \; \mathrm{type} -> Γ,x:AB(x)type\Gamma, x:A \vdash B(x) \; \mathrm{type}

    F. Schaefer

    diff, v36, current