From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: X-Spam-Checker-Version: SpamAssassin 3.4.2 (2018-09-13) on inbox.vuxu.org X-Spam-Level: X-Spam-Status: No, score=-0.9 required=5.0 tests=DKIMWL_WL_MED,DKIM_SIGNED, DKIM_VALID,DKIM_VALID_EF,HEADER_FROM_DIFFERENT_DOMAINS, MAILING_LIST_MULTI,RCVD_IN_DNSWL_NONE autolearn=ham autolearn_force=no version=3.4.2 Received: from mail-pf1-x43a.google.com (mail-pf1-x43a.google.com [IPv6:2607:f8b0:4864:20::43a]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id 41da1d3a for ; Sun, 11 Aug 2019 16:55:41 +0000 (UTC) Received: by mail-pf1-x43a.google.com with SMTP id 22sf1024365pfn.22 for ; Sun, 11 Aug 2019 09:55:40 -0700 (PDT) ARC-Seal: i=2; a=rsa-sha256; t=1565542539; cv=pass; d=google.com; s=arc-20160816; b=j/NwnIWWOmTfpCfZ9fTwlpu89VpUIp70A8cSq9iPjfp5d9Vb+Qko2qfR2bLVsb/FKQ mtB/ZrhEszF5BsBlp8cFSicKpjJCCuYb+KC+Srq//yBCrMQ+ca/fyrUwu8elsN/kwZjq go7BVZ6KQRT6SacKo20qcWPgLRnS9VICjGhGGC9rVxmbyykuu9OfTueo9YhjzbLvsdMD /ItJi5uZR0ccfi1wGC2KWg86odz7qhmL3HJYR6ld4GSOjPJv5Sd5DUOFvh66oR6dvT2G PqlQgKTUo7KHNIuAf3ASHJql0wVNoUqz/HE2kWQIEAScDZDM7wsMRmymuldlBC92dmjl t7Sw== ARC-Message-Signature: i=2; a=rsa-sha256; c=relaxed/relaxed; d=google.com; s=arc-20160816; h=list-unsubscribe:list-archive:list-help:list-post:list-id :mailing-list:precedence:content-transfer-encoding:to:subject :message-id:date:from:in-reply-to:references:mime-version:sender :dkim-signature; bh=faqmUts83sTdHSjEspFDwkutx6LJE4OxAjEx9IWYDBo=; b=ekeRUQ7ACagJCn+sun+Pgy4FrLYMW66DZqIG0BIQJurQie3gDz+xdWIUXR9SCPkez6 OkZoflg8IJjkqEFDdOlNqpkVYvzHh/F34plfoSJviX36ODDxlX+Fk9zogiF+jsDNB/kO IRjHa7EubD/2e/5t6nl9iqyRHJFExzitZ0y4gBS8VFZZTBD7NwdND474kA3Tb0b5+Wl9 /NlVqtPYou0AmZx8zEE9lOUHfqzKrsW0mA9IMyOTfPoBhxw9GCaLMNOkqGvrnmr+lUZG iMpsj6SvFn4iBIv6MDbxKvfrm3Xuxlcp3/tLKqp7WBjCa1htcR9u/R60Yj7WFXaXezuD frLQ== ARC-Authentication-Results: i=2; gmr-mx.google.com; dkim=pass header.i=@sandiego-edu.20150623.gappssmtp.com header.s=20150623 header.b="ypvFP/1z"; spf=pass (google.com: domain of shulman@sandiego.edu designates 2607:f8b0:4864:20::b41 as permitted sender) smtp.mailfrom=shulman@sandiego.edu DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=20161025; h=sender:mime-version:references:in-reply-to:from:date:message-id :subject:to:content-transfer-encoding:x-original-sender :x-original-authentication-results:precedence:mailing-list:list-id :list-post:list-help:list-archive:list-unsubscribe; bh=faqmUts83sTdHSjEspFDwkutx6LJE4OxAjEx9IWYDBo=; b=G4T/9otTqjoE0x9xPEgiVOSX5H5pEOJBxvwGmANIy8tYMcupZLpndgXuQCotFBFvM7 RIUJBDoNDZnoEyVSdMCwHHXbaBaWxVBopCkZt/+msZmc9LyjNnH/T8I7XrLh0htU7KfS FEgBDYjkzUX2C57KFnze9zfF0sz69cyNJUnhVuQhEG3tFccWeLcbFGRvugtYsue8S+IK k3wlS6sy+hK+yfTfVVlGbVN3n0g+dqYutHz0IeQAmznKFNr0TobT726Wf54rwIH1xiU7 +0f5ISZEOtNb0x++v99qIjqoTl4WlEvlhIHVVBpXKvvBxzfVLBPtGY2Vdym6kC7IRB9w i9cw== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=sender:x-gm-message-state:mime-version:references:in-reply-to:from :date:message-id:subject:to:content-transfer-encoding :x-original-sender:x-original-authentication-results:precedence :mailing-list:list-id:x-spam-checked-in-group:list-post:list-help :list-archive:list-unsubscribe; bh=faqmUts83sTdHSjEspFDwkutx6LJE4OxAjEx9IWYDBo=; b=Y2GKc+bRNW8yyaUIIp059pBMDgLgtRzsIKaTa1Ymw8uw+c1xF+l1IaB1PFqV2LkZt9 U6/fgvKyWHG+E/jPizeb5PQb00+f6YomOCPyq5zn+hJaGSC4Fn8JBb5fHC5dvXtEr5Ui yNvK9gWruJH8LGM/MOTc5CM8QIdu5pcti4uMNPFSDeh3cryfXReltPrfVPKfEe5Rs+7+ TFNVXmd2VkJONEZxENZS329qgAZkrfvx67F4a4ZSX4rWX6alaWK3fYRiUGOofmwTqAOi GfVaB64vOduAvpq8HPKwcQrrH41A7SOMLEZATFXRyoC/tyR9WbAyLDa3J5e48YGvgiYa zydQ== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APjAAAXE41VdkOiZEXNmRJ+5DjX7qtfB+HztRtT8xPa76iaAghAXg8eu ls9u+CQAomNGseHvDwsdY7A= X-Google-Smtp-Source: APXvYqxn/wTapHQMb2HKFt+hW8vnSoCkWbeD0CYMKDCdSInne4i10lBdYbp/aXNZ+VZ1+Sz6D9KscA== X-Received: by 2002:a17:90a:be15:: with SMTP id a21mr12748132pjs.74.1565542539196; Sun, 11 Aug 2019 09:55:39 -0700 (PDT) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:a63:954c:: with SMTP id t12ls22443672pgn.12.gmail; Sun, 11 Aug 2019 09:55:38 -0700 (PDT) X-Received: by 2002:a63:3203:: with SMTP id y3mr27019601pgy.191.1565542538698; Sun, 11 Aug 2019 09:55:38 -0700 (PDT) ARC-Seal: i=1; a=rsa-sha256; t=1565542538; cv=none; d=google.com; s=arc-20160816; b=oxsmshmYuvRpI7H7sU5T7TJeWZmzVWBnltC6uiC9O/KqgSLr0Ccs4uH7mXs0WDoVHG iheW2/8LhyDbfooNE5ozbce43zfffkzbSBdR8EbwwGimdOwVzcbHSftnwojpZ+lFL7ls ZV9nw2+eLEUNPyFrL3tIOqEDgGaQ8+nEs6r+94FANUM81sKvbwAlaLp54mmtBJRt61TX V/Mg1YH1Eva/rzYels/rmPOIVFUk0d8cx6Id/8CuED7C4hvROzTtKXagjbNhj2ET2NzM p7ENx6dCWQWnLkN0SlMDtwdS73x0cUiEuDf53UMKwbZaHL7s5wqt0zS6/AqvGLvUGSwc n9Vg== ARC-Message-Signature: i=1; a=rsa-sha256; c=relaxed/relaxed; d=google.com; s=arc-20160816; h=content-transfer-encoding:to:subject:message-id:date:from :in-reply-to:references:mime-version:dkim-signature; bh=MqAOuE3dHz5gYiIuwVdXe4dd/KfemEvc/+WPiJVXuuM=; b=VvWHepV5aG7IISIa+TzDIZ3bOya5o69WB3jecbv5Xck49avorRquharo1pn0QTAeoc 3WDW9cmXhbnWAKH2vSzUfcqKtxqemoYl/35+WrjRi+dHQwNrNR/MKCdNM0AxrwHtJY5b hS77QTshQptuKEANgAtTpAfhIX6c28s0Iny0dvqZA5+xm7xqKNDRXaHJkskJz8bADK1h /iJ5Nk0s2gSNsI72GWiSEDG/Lr8AvOHZoYtcB1ImxgO6gQmU9F8g/m9oDP3UDbrSWiZZ 9SMqot8erfg2ckifEE9zWOi+Yed8IKjf1VxgjTPLFNRFMK7GCfiWgBdXdYINpi5sRr+H oNaQ== ARC-Authentication-Results: i=1; gmr-mx.google.com; dkim=pass header.i=@sandiego-edu.20150623.gappssmtp.com header.s=20150623 header.b="ypvFP/1z"; spf=pass (google.com: domain of shulman@sandiego.edu designates 2607:f8b0:4864:20::b41 as permitted sender) smtp.mailfrom=shulman@sandiego.edu Received: from mail-yb1-xb41.google.com (mail-yb1-xb41.google.com. [2607:f8b0:4864:20::b41]) by gmr-mx.google.com with ESMTPS id b24si349692pjq.1.2019.08.11.09.55.38 for (version=TLS1_3 cipher=AEAD-AES128-GCM-SHA256 bits=128/128); Sun, 11 Aug 2019 09:55:38 -0700 (PDT) Received-SPF: pass (google.com: domain of shulman@sandiego.edu designates 2607:f8b0:4864:20::b41 as permitted sender) client-ip=2607:f8b0:4864:20::b41; Received: by mail-yb1-xb41.google.com with SMTP id j199so38130622ybg.5 for ; Sun, 11 Aug 2019 09:55:38 -0700 (PDT) X-Received: by 2002:a25:7005:: with SMTP id l5mr22655478ybc.452.1565542537682; Sun, 11 Aug 2019 09:55:37 -0700 (PDT) Received: from mail-yw1-f42.google.com (mail-yw1-f42.google.com. [209.85.161.42]) by smtp.gmail.com with ESMTPSA id i137sm23183963ywa.3.2019.08.11.09.55.37 for (version=TLS1_3 cipher=AEAD-AES128-GCM-SHA256 bits=128/128); Sun, 11 Aug 2019 09:55:37 -0700 (PDT) Received: by mail-yw1-f42.google.com with SMTP id g19so37311556ywe.2 for ; Sun, 11 Aug 2019 09:55:37 -0700 (PDT) X-Received: by 2002:a81:6286:: with SMTP id w128mr1869407ywb.272.1565542536778; Sun, 11 Aug 2019 09:55:36 -0700 (PDT) MIME-Version: 1.0 References: <9d23061c-4b7a-4d69-9c22-f28261ad3b33@googlegroups.com> <06e24c98-7409-4e75-88ee-a6e1bb891e1e@www.fastmail.com> In-Reply-To: From: Michael Shulman Date: Sun, 11 Aug 2019 12:55:24 -0400 X-Gmail-Original-Message-ID: Message-ID: Subject: Re: [HoTT] New theorem prover Arend is released To: "HomotopyTypeTheory@googlegroups.com" Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable X-Original-Sender: shulman@sandiego.edu X-Original-Authentication-Results: gmr-mx.google.com; dkim=pass header.i=@sandiego-edu.20150623.gappssmtp.com header.s=20150623 header.b="ypvFP/1z"; spf=pass (google.com: domain of shulman@sandiego.edu designates 2607:f8b0:4864:20::b41 as permitted sender) smtp.mailfrom=shulman@sandiego.edu Precedence: list Mailing-list: list HomotopyTypeTheory@googlegroups.com; contact HomotopyTypeTheory+owners@googlegroups.com List-ID: X-Google-Group-Id: 1041266174716 List-Post: , List-Help: , List-Archive: , In an off-list discussion, Valery clarified to me that \use\level for \func does not actually put the type in the corresponding universe, only marks it as having the corresponding homotopy level (and added some documentation https://arend-lang.github.io/documentation/language-reference/definitions/l= evel#use-level-for-functions). However, it also emerged that F(A,p) can also be *defined* for an arbitrary type A (and an externally fixed n) as \record F (A : \Type) (p : ofHLevel A n) { | in : A } \where \use \level isntype (A : \Type) (p : ofHLevel A n) : ofHLevel A n = =3D> p and that because records have definitional eta-conversion, this F(A,p) will actually be definitionally isomorphic to A, rather than just equivalent to it. In particular, since \Prop is invariant under predicative level, this means that (in the current version of Arend) every h-proposition is definitionally isomorphic to one living in the lowest universe. This is not quite as strong as Voeovdsky's propositional resizing rule (that every h-proposition *itself* lives in the lowest universe), but I find it about equally semantically dubious, and in particular it's unclear whether it has any univalent models (not even classically in simplicial sets). Perhaps it does; but until such models are known, it may be better to avoid using this feature for \Prop, and perhaps add a restriction to Arend preventing it. On Sun, Aug 11, 2019 at 8:39 AM Michael Shulman wrot= e: > > On Sun, Aug 11, 2019 at 6:47 AM Valery Isaev wro= te: > > I don't remember well, but I think the idea is that you need to prove t= hat there is a trivial cofibration Eq(A,B) -> F(U^I,A,B), where the first o= bject is the object of equivalences between A and B and the second object i= s the fiber of U^I over A and B. The fact that this map is a weak equivalen= ce is just the univalence axiom. The problem is to show that it is a cofibr= ation and whether this is true or not depends on the definition of Eq(A,B).= I don't actually remember whether I finished this proof. > > Yes, in a straightforward approach that's what you would need to > prove, or more precisely that the map Eq -> U^I is a trivial > cofibration in the slice over UxU (since the lifting has to be done in > the universal case rather than just over each global element). > Possibly you could get away with less, since you're only asking to > recover the action of the given equivalence as a function (rather than > the entire equivalence data). Perhaps a clever choice of fibration > structure in the equivalence extension property would suffice. > > > Since F(A,p) is the usual (inductive) data type, you can do everything = you can do with other data types. In particular, since it has only one cons= tructor with one parameter A, it is easy to proof that it is equivalent to = A. > > Okay, it sounds like you're saying that the datatype defined with > \use\level is F(A,p) and the analogous one defined without \use\level > is A, while both have their usual rules but are not identical. In > this case you have somewhat *more* than an equivalence between F(A,p) > and A, at least if all datatypes have their usual definitional > computation rules, since an arbitrary type "F(A,p)" that is only > *equivalent* to a datatype will only inherit a typal computation rule. > It seems plausible though that one could construct universes of > n-types that are strictly closed under a given datatype construction > that preserves n-types (at least, assuming one can do the same for > arbitrary universes). > > However, the documentation also suggests that \use\level can be > applied to plain definitions (\func). How is F(A,p) distinguished > from A in this case? --=20 You received this message because you are subscribed to the Google Groups "= Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/= HomotopyTypeTheory/CAOvivQzjOnpZj27f44GyDjtX1-WNwWN8aqOnv3L7gkVnpYpnyw%40ma= il.gmail.com.