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  1. the concept of a weak type theory as a dependent type theory where all type formers are weak in the sense that they use identity types instead of judgmental equality in the computation and uniqueness rules

    Anonymouse

    v1, current

  2. The van der Berg paper was not the first paper which talks about weak type theory. Theo Winterhalter’s thesis talks about weak type theory was published in 2020 before the van der Berg paper was put up on the arXiv in 2021.

    Anonymouse

    diff, v6, current

  3. Just a note: both “propositional type theory” and “weak type theory” are synonyms of “objective type theory” in this subfield. See e.g. van der Berg’s slides at the DutchCATS conference on weak type theory:

    Propositional type theory is a version of type theory without definitional equality and in which all computation rules are stated in propositional form. (Other names: homotopy type theory with explicit conversions or objective type theory.)

    Or Spadetto’s slides from the same conference:

    Objective/propositional/weak type theory

    In:

    Benno van den Berg, Martijn den Besten, Quadratic type checking for objective type theory, 2021.

    All the existing literature, starting from Winterhalter’s paper and continuing to the present day, imply that “weak type theory” does not have any definitional equality / conversions / judgmental equality, and this includes conversion of types to defined aliases. Even explicit substitution uses identity types instead of judgmental equality.

    So this article should be merged into objective type theory.

    • CommentRowNumber4.
    • CommentAuthorP
    • CommentTimeMay 17th 2025

    renamed page to be about any type theory with propositional computation rules, and moved stuff about weak type theory over to objective type theory

    diff, v7, current

    • CommentRowNumber5.
    • CommentAuthorP
    • CommentTimeMay 17th 2025

    simpler name

    diff, v7, current

    • CommentRowNumber6.
    • CommentAuthorP
    • CommentTimeMay 17th 2025

    Actually, changed my mind. The authors never really address the issue of definitions in the literature. The thing is that “propositional type theory” is overloaded as a term: in van der Berg, judgmental equality is α-equivalence and definitions are made with identity types, but in Spadetto, the dependent type theory comes with judgmental definitional equality.

    diff, v8, current