Discussion of Homotopy Type Theory and Univalent Foundations
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From: Gabriel Scherer <gabriel.scherer@gmail.com>
To: Nicolai Kraus <nicolai.kraus@gmail.com>
Cc: Felix Rech <s9ferech@gmail.com>,
	 Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Re: Why do we need judgmental equality?
Date: Sat, 9 Feb 2019 15:26:54 +0100	[thread overview]
Message-ID: <CAPFanBGNoL+AKXWh2a8yQ2C2N3bF0BZjgijmqQdkHJ77oy_rrg@mail.gmail.com> (raw)
In-Reply-To: <CA+AZBBpfg5RcKuLAF9YXceabb5JKsHw2LKsT-Ra+xyuCJN2nPw@mail.gmail.com>

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A relevant citation for this line of thinking (if we reduce implicit
computation, can we gain usability some other way?) is the work on the
Zombie programming language, which tried to weaken beta-reduction in a
dependent language (restricting it to values only, as in F*) to obtain a
more automatic form of reasoning upto congruence (definitional equalities
can be put in the context and used from there).

  Programming up to Congruence
  Vilhelm Sjöberg and Stephanie Weirich, 2015
  http://www.cs.yale.edu/homes/vilhelm/papers/popl15congruence.pdf


On Sat, Feb 9, 2019 at 3:05 PM Nicolai Kraus <nicolai.kraus@gmail.com>
wrote:

> On Sat, Feb 9, 2019 at 11:53 AM Felix Rech <s9ferech@gmail.com> wrote:
>
>> One of the motivations for my question was that I actually expect
>> usability benefits if one worked in a dependent type theory without
>> judgmental equality that has good support by a proof assistant.
>>
>
> Yes, me too (but I think *a lot* of work needs to be done before we can
> have a proof assistant based on this idea which provides better usability
> than the current ones).
> I agree with your points. But I think "x + 0 = x versus 0 + x = x" is an
> example where I'd expect that one should be able to produce a
> conservativity proof and use both at the same time instead of choosing one.
> I think it's not necessary that we restrict ourselves to computation rules
> that come from actual definitions; anything that is "constructively
> conservative" over a weak theory should be allowed. In Agda, one can have
> additional computation rules, but it's not a safe feature.
> Nicolai
>
>
>
>>    1. Judgmental equality cannot be taken as assumptions. If one wants
>>    to use judgmental equalities then one has to give concrete definitions that
>>    satisfy those equalities and cannot hide the definition details. This makes
>>    it impossible to change definitions later on without breaking constructions
>>    that depend on them.
>>    2. In nontrivial definitions, judgmental equalities seem more
>>    arbitrary than natural. If we define addition of natural numbers for
>>    example then we can choose between x + 0 = x and 0 + x = x as judgmental
>>    equality but we cannot have both. This makes it hard to find the right
>>    definitions and to predict their behavior.
>>
>> Another motivation was of course that it would simplify the
>> implementation of proof checkers and parts of the metatheory.
>>
>> I would appreciate any comments on this.
>>
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  reply	other threads:[~2019-02-09 14:16 UTC|newest]

Thread overview: 71+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-01-30 11:54 [HoTT] " Felix Rech
2019-02-05 23:00 ` [HoTT] " Matt Oliveri
2019-02-06  4:13   ` Anders Mörtberg
2019-02-09 11:55     ` Felix Rech
2019-02-16 15:59     ` Thorsten Altenkirch
2019-02-17  1:25       ` Michael Shulman
2019-02-17  7:56         ` Thorsten Altenkirch
2019-02-17  9:14           ` Matt Oliveri
2019-02-17  9:18           ` Michael Shulman
2019-02-17 10:52             ` Thorsten Altenkirch
2019-02-17 11:35               ` streicher
2019-02-17 11:44                 ` Thorsten Altenkirch
2019-02-17 14:24                   ` Bas Spitters
2019-02-17 19:36                   ` Thomas Streicher
2019-02-17 21:41                     ` Thorsten Altenkirch
2019-02-17 12:08             ` Matt Oliveri
2019-02-17 12:13               ` Matt Oliveri
2019-02-20  0:22               ` Michael Shulman
2019-02-17 14:22           ` [Agda] " Andreas Abel
2019-02-17  9:05         ` Matt Oliveri
2019-02-17 13:29         ` Nicolai Kraus
2019-02-08 21:19 ` Martín Hötzel Escardó
2019-02-08 23:31   ` Valery Isaev
2019-02-09  1:41     ` Nicolai Kraus
2019-02-09  8:04       ` Valery Isaev
2019-02-09  1:58     ` Jon Sterling
2019-02-09  8:16       ` Valery Isaev
2019-02-09  1:30   ` Nicolai Kraus
2019-02-09 11:38   ` Thomas Streicher
2019-02-09 13:29     ` Thorsten Altenkirch
2019-02-09 13:40       ` Théo Winterhalter
2019-02-09 11:57   ` Felix Rech
2019-02-09 12:39     ` Martín Hötzel Escardó
2019-02-11  6:58     ` Matt Oliveri
2019-02-18 17:37   ` Martín Hötzel Escardó
2019-02-18 19:22     ` Licata, Dan
2019-02-18 20:23       ` Martín Hötzel Escardó
2019-02-09 11:53 ` Felix Rech
2019-02-09 14:04   ` Nicolai Kraus
2019-02-09 14:26     ` Gabriel Scherer [this message]
2019-02-09 14:44     ` Jon Sterling
2019-02-09 20:34       ` Michael Shulman
2019-02-11 12:17         ` Matt Oliveri
2019-02-11 13:04           ` Michael Shulman
2019-02-11 15:09             ` Matt Oliveri
2019-02-11 17:20               ` Michael Shulman
2019-02-11 18:17                 ` Thorsten Altenkirch
2019-02-11 18:45                   ` Alexander Kurz
2019-02-11 22:58                     ` Thorsten Altenkirch
2019-02-12  2:09                       ` Jacques Carette
2019-02-12 11:03                   ` Matt Oliveri
2019-02-12 15:36                     ` Thorsten Altenkirch
2019-02-12 15:59                       ` Matt Oliveri
2019-02-11 19:27                 ` Matt Oliveri
2019-02-11 21:49                   ` Michael Shulman
2019-02-12  9:01                     ` Matt Oliveri
2019-02-12 17:54                       ` Michael Shulman
2019-02-13  6:37                         ` Matt Oliveri
2019-02-13 10:01                           ` Ansten Mørch Klev
2019-02-11 20:11                 ` Matt Oliveri
2019-02-11  8:23       ` Matt Oliveri
2019-02-11 13:03         ` Jon Sterling
2019-02-11 13:22           ` Matt Oliveri
2019-02-11 13:37             ` Jon Sterling
2019-02-11  6:51   ` Matt Oliveri
2019-02-09 12:30 ` [HoTT] " Thorsten Altenkirch
2019-02-11  7:01   ` Matt Oliveri
2019-02-11  8:04     ` Valery Isaev
2019-02-11  8:28       ` Matt Oliveri
2019-02-11  8:37         ` Matt Oliveri
2019-02-11  9:32           ` Rafaël Bocquet

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