TheHoTTGame should be more usable now!
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jlh
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgcb7a42f">Planning The HoTT Game</a>
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<ul>
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<li><a href="#org86f8abf">Aims of the HoTT Game</a></li>
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<li><a href="#org891bd67">Barriers</a></li>
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<li><a href="#org865979d">Format</a></li>
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<li><a href="#org7cb3a5d">Content</a></li>
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<li><a href="#org313668c">Debriefs</a></li>
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</li>
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<div id="outline-container-orgcb7a42f" class="outline-2">
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<h2 id="orgcb7a42f">Planning The HoTT Game</h2>
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<div id="outline-container-org86f8abf" class="outline-3">
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<h3 id="org86f8abf">Aims of the HoTT Game</h3>
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<div class="outline-text-3" id="text-org86f8abf">
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<ul class="org-ul">
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<li>To get mathematicians with no experience in proof verification interested in HoTT and able to use Agda for HoTT</li>
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<li>[?] Work towards showing an interesting result in HoTT</li>
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<li>Try to balance hiding cubical implementations whilst exploiting their advantages</li>
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</ul>
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</div>
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<h3 id="org891bd67">Barriers</h3>
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<div class="outline-text-3" id="text-org891bd67">
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<ul class="org-ul">
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<li>HOLD Installation of emacs</li>
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<li>TODO Usage of emacs</li>
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<li>TODO General type theoretic foundations</li>
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<li>TODO Cubical type theory</li>
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</ul>
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</div>
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</div>
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<h3 id="org865979d">Format</h3>
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<ul class="org-ul">
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<li>[?] Everything done in .agda files</li>
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<li>Partially written code with spaces for participants to fill in + answer files</li>
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<li>Levels set out with mini-bosses like in Nat Num Game, but with an overall boss</li>
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<li>[?] Side quests</li>
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<li>References to Harper lectures and HoTT book</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org7cb3a5d" class="outline-3">
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<h3 id="org7cb3a5d">Content</h3>
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<div class="outline-text-3" id="text-org7cb3a5d">
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<ul class="org-ul">
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<li>emacs usage</li>
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<li>agda usage
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<ul class="org-ul">
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|
||||||
<li>basic commands (all covered in <a href="https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html">https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html</a>)</li>
|
|
||||||
<li>recommend doom emacs</li>
|
|
||||||
<li>implicit/explicit arguments</li>
|
|
||||||
<li>holes and inferred types</li>
|
|
||||||
<li><code>_+_</code> vs <code>plus__</code></li>
|
|
||||||
</ul></li>
|
|
||||||
<li>type theory basics
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>meta (judgemental/definitional) equality vs internal (propositional) equality
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>function extensionality</li>
|
|
||||||
</ul></li>
|
|
||||||
<li>type formation
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>inductive types
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>(side Q) positive and negative constructions of Pi/Sigma types</li>
|
|
||||||
<li><code>data</code> and <code>record</code></li>
|
|
||||||
</ul></li>
|
|
||||||
</ul></li>
|
|
||||||
<li>universes</li>
|
|
||||||
<li>recursors / pattern matching</li>
|
|
||||||
<li>(side Q) some natural number exercises as early evidence of being able to ’do maths’?</li>
|
|
||||||
<li>different notions of equivalence
|
|
||||||
<ol class="org-ol">
|
|
||||||
<li>fibers contractable</li>
|
|
||||||
<li>quasi-inverse</li>
|
|
||||||
<li>zig-zag</li>
|
|
||||||
</ol></li>
|
|
||||||
<li>(side Q) types are infinity groupoids</li>
|
|
||||||
<li>extra paths (univalence, fun ext, HITs)</li>
|
|
||||||
</ul></li>
|
|
||||||
<li>HoTT
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>basics
|
|
||||||
<ol class="org-ol">
|
|
||||||
<li>meta interval, identity type vs path type
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>mention identity type for compatability with other sources, but just use path type</li>
|
|
||||||
</ul></li>
|
|
||||||
<li>path type on other types</li>
|
|
||||||
<li>dependent path type PathP vs path over</li>
|
|
||||||
<li>univalence</li>
|
|
||||||
<li>the (non)-issue of J in (Cu)TT</li>
|
|
||||||
<li>isContr, isProp, isSet</li>
|
|
||||||
<li>drawing pictures</li>
|
|
||||||
</ol></li>
|
|
||||||
<li>Structures, using univalence to transport
|
|
||||||
<ol class="org-ol">
|
|
||||||
<li>transporting results between isomorphic structures</li>
|
|
||||||
</ol></li>
|
|
||||||
<li>HITs, examples
|
|
||||||
<ol class="org-ol">
|
|
||||||
<li>the constructed interval</li>
|
|
||||||
<li>booleans and covers</li>
|
|
||||||
<li>S<sup>n</sup></li>
|
|
||||||
<li>S<sup>1</sup> with 2 cw structures equiv</li>
|
|
||||||
</ol></li>
|
|
||||||
<li>Homotopy n-types
|
|
||||||
<ol class="org-ol">
|
|
||||||
<li>homotopy levels being closed under type constructions, in particular Set and ETT inside HoTT
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>in particular sigma types</li>
|
|
||||||
</ul></li>
|
|
||||||
</ol></li>
|
|
||||||
</ul></li>
|
|
||||||
</ul>
|
|
||||||
</div>
|
|
||||||
</div>
|
|
||||||
|
|
||||||
<div id="outline-container-org313668c" class="outline-3">
|
|
||||||
<h3 id="org313668c">Debriefs</h3>
|
|
||||||
<div class="outline-text-3" id="text-org313668c">
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>2021 July 15; Homotopy n-types
|
|
||||||
<ul class="org-ul">
|
|
||||||
<li>watched (Harper) lecture 15 on Sets being closed under type formations ->- motivates showing in Agda Sets closed under Sigma.</li>
|
|
||||||
<li>Harper does product case, claiming sigma case follows analogously,</li>
|
|
||||||
<li>attempt proof in Cubical Agda but highly non-obvious how to use that fibers are Sets.</li>
|
|
||||||
<li>difficulty is that PathP not in one fiber, but PathOver is, AND PathOver <-> PathP NON-obvious</li>
|
|
||||||
<li>Easy to generalize situation to n-types being closed under Sigma (7.1.8 in HoTT book), we showed this assuming PathPIsoPath</li>
|
|
||||||
</ul></li>
|
|
||||||
</ul>
|
|
||||||
</div>
|
|
||||||
</div>
|
|
||||||
</div>
|
|
||||||
</div>
|
|
||||||
<div id="postamble" class="status">
|
|
||||||
<p class="author">Author: JLH KL</p>
|
|
||||||
<p class="date">Created: 2021-07-19 Mon 18:31</p>
|
|
||||||
</div>
|
|
||||||
</body>
|
|
||||||
</html>
|
|
||||||
107
Plan.md
107
Plan.md
@ -1,107 +0,0 @@
|
|||||||
|
|
||||||
# Table of Contents
|
|
||||||
|
|
||||||
- [Planning The HoTT Game](#org3bb90ed)
|
|
||||||
- [Aims of the HoTT Game](#orga8d795d)
|
|
||||||
- [Barriers](#org122a1d0)
|
|
||||||
- [Format](#org3ea389f)
|
|
||||||
- [Content](#org70d2231)
|
|
||||||
- [Debriefs](#org37fbeb9)
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
<a id="org3bb90ed"></a>
|
|
||||||
|
|
||||||
# Planning The HoTT Game
|
|
||||||
|
|
||||||
|
|
||||||
<a id="orga8d795d"></a>
|
|
||||||
|
|
||||||
## Aims of the HoTT Game
|
|
||||||
|
|
||||||
- To get mathematicians with no experience in proof verification interested in HoTT and able to use Agda for HoTT
|
|
||||||
- [?] Work towards showing an interesting result in HoTT
|
|
||||||
- Try to balance hiding cubical implementations whilst exploiting their advantages
|
|
||||||
|
|
||||||
|
|
||||||
<a id="org122a1d0"></a>
|
|
||||||
|
|
||||||
## Barriers
|
|
||||||
|
|
||||||
- HOLD Installation of emacs
|
|
||||||
- TODO Usage of emacs
|
|
||||||
- TODO General type theoretic foundations
|
|
||||||
- TODO Cubical type theory
|
|
||||||
|
|
||||||
|
|
||||||
<a id="org3ea389f"></a>
|
|
||||||
|
|
||||||
## Format
|
|
||||||
|
|
||||||
- [?] Everything done in .agda files
|
|
||||||
- Partially written code with spaces for participants to fill in + answer files
|
|
||||||
- Levels set out with mini-bosses like in Nat Num Game, but with an overall boss
|
|
||||||
- [?] Side quests
|
|
||||||
- References to Harper lectures and HoTT book
|
|
||||||
|
|
||||||
|
|
||||||
<a id="org70d2231"></a>
|
|
||||||
|
|
||||||
## Content
|
|
||||||
|
|
||||||
- emacs usage
|
|
||||||
- agda usage
|
|
||||||
- basic commands (all covered in <https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html>)
|
|
||||||
- recommend doom emacs
|
|
||||||
- implicit/explicit arguments
|
|
||||||
- holes and inferred types
|
|
||||||
- `_+_` vs `plus__`
|
|
||||||
- type theory basics
|
|
||||||
- meta (judgemental/definitional) equality vs internal (propositional) equality
|
|
||||||
- function extensionality
|
|
||||||
- type formation
|
|
||||||
- inductive types
|
|
||||||
- (side Q) positive and negative constructions of Pi/Sigma types
|
|
||||||
- `data` and `record`
|
|
||||||
- universes
|
|
||||||
- recursors / pattern matching
|
|
||||||
- (side Q) some natural number exercises as early evidence of being able to ’do maths’?
|
|
||||||
- different notions of equivalence
|
|
||||||
1. fibers contractable
|
|
||||||
2. quasi-inverse
|
|
||||||
3. zig-zag
|
|
||||||
- (side Q) types are infinity groupoids
|
|
||||||
- extra paths (univalence, fun ext, HITs)
|
|
||||||
- HoTT
|
|
||||||
- basics
|
|
||||||
1. meta interval, identity type vs path type
|
|
||||||
- mention identity type for compatability with other sources, but just use path type
|
|
||||||
2. path type on other types
|
|
||||||
3. dependent path type PathP vs path over
|
|
||||||
4. univalence
|
|
||||||
5. the (non)-issue of J in (Cu)TT
|
|
||||||
6. isContr, isProp, isSet
|
|
||||||
7. drawing pictures
|
|
||||||
- Structures, using univalence to transport
|
|
||||||
1. transporting results between isomorphic structures
|
|
||||||
- HITs, examples
|
|
||||||
1. the constructed interval
|
|
||||||
2. booleans and covers
|
|
||||||
3. S<sup>n</sup>
|
|
||||||
4. S<sup>1</sup> with 2 cw structures equiv
|
|
||||||
- Homotopy n-types
|
|
||||||
1. homotopy levels being closed under type constructions, in particular Set and ETT inside HoTT
|
|
||||||
- in particular sigma types
|
|
||||||
|
|
||||||
|
|
||||||
<a id="org37fbeb9"></a>
|
|
||||||
|
|
||||||
## Debriefs
|
|
||||||
|
|
||||||
- 2021 July 15; Homotopy n-types
|
|
||||||
- watched (Harper) lecture 15 on Sets being closed under type formations ->- motivates showing in Agda Sets closed under Sigma.
|
|
||||||
- Harper does product case, claiming sigma case follows analogously,
|
|
||||||
- attempt proof in Cubical Agda but highly non-obvious how to use that fibers are Sets.
|
|
||||||
- difficulty is that PathP not in one fiber, but PathOver is, AND PathOver <-> PathP NON-obvious
|
|
||||||
- Easy to generalize situation to n-types being closed under Sigma (7.1.8 in HoTT book), we showed this assuming PathPIsoPath
|
|
||||||
|
|
||||||
84
Plan.org
84
Plan.org
@ -27,52 +27,52 @@
|
|||||||
|
|
||||||
- emacs usage
|
- emacs usage
|
||||||
- agda usage
|
- agda usage
|
||||||
- basic commands (all covered in https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html)
|
+ basic commands (all covered in https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html)
|
||||||
- recommend doom emacs
|
+ recommend doom emacs
|
||||||
- implicit/explicit arguments
|
+ implicit/explicit arguments
|
||||||
- holes and inferred types
|
+ holes and inferred types
|
||||||
- =_+_= vs ~plus__~
|
+ =_+_= vs =plus__=
|
||||||
- type theory basics
|
- type theory basics
|
||||||
- meta (judgemental/definitional) equality vs internal (propositional) equality
|
+ meta (judgemental/definitional) equality vs internal (propositional) equality
|
||||||
- function extensionality
|
- function extensionality
|
||||||
- type formation
|
+ type formation
|
||||||
- inductive types
|
- inductive types
|
||||||
- (side Q) positive and negative constructions of Pi/Sigma types
|
+ (side Q) positive and negative constructions of Pi/Sigma types
|
||||||
- ~data~ and ~record~
|
+ =data= and =record=
|
||||||
- universes
|
+ universes
|
||||||
- recursors / pattern matching
|
+ recursors / pattern matching
|
||||||
- (side Q) some natural number exercises as early evidence of being able to 'do maths'?
|
+ (side Q) some natural number exercises as early evidence of being able to 'do maths'?
|
||||||
- different notions of equivalence
|
+ different notions of equivalence
|
||||||
a) fibers contractable
|
- fibers contractable
|
||||||
b) quasi-inverse
|
- quasi-inverse
|
||||||
c) zig-zag
|
- zig-zag
|
||||||
- (side Q) types are infinity groupoids
|
+ (side Q) types are infinity groupoids
|
||||||
- extra paths (univalence, fun ext, HITs)
|
+ extra paths (univalence, fun ext, HITs)
|
||||||
- HoTT
|
- HoTT
|
||||||
- basics
|
+ basics
|
||||||
a) meta interval, identity type vs path type
|
- meta interval, identity type vs path type
|
||||||
- mention identity type for compatability with other sources, but just use path type
|
+ mention identity type for compatability with other sources, but just use path type
|
||||||
b) path type on other types
|
- path type on other types
|
||||||
c) dependent path type PathP vs path over
|
- dependent path type PathP vs path over
|
||||||
d) univalence
|
- univalence
|
||||||
e) the (non)-issue of J in (Cu)TT
|
- the (non)-issue of J in (Cu)TT
|
||||||
f) isContr, isProp, isSet
|
- =isContr, isProp, isSet=
|
||||||
g) drawing pictures
|
- drawing pictures
|
||||||
- Structures, using univalence to transport
|
+ Structures, using univalence to transport
|
||||||
a) transporting results between isomorphic structures
|
- transporting results between isomorphic structures
|
||||||
- HITs, examples
|
+ HITs, examples
|
||||||
a) the constructed interval
|
- the constructed interval
|
||||||
b) booleans and covers
|
- booleans and covers
|
||||||
c) S^n
|
- S^n
|
||||||
d) S^1 with 2 cw structures equiv
|
- S^1 with 2 cw structures equiv
|
||||||
- Homotopy n-types
|
+ Homotopy n-types
|
||||||
a) homotopy levels being closed under type constructions, in particular Set and ETT inside HoTT
|
- homotopy levels being closed under type constructions, in particular Set and ETT inside HoTT
|
||||||
* in particular sigma types
|
+ in particular sigma types
|
||||||
|
|
||||||
** Debriefs
|
** Debriefs
|
||||||
- 2021 July 15; Homotopy n-types
|
- 2021 July 15; Homotopy n-types
|
||||||
- watched (Harper) lecture 15 on Sets being closed under type formations ->- motivates showing in Agda Sets closed under Sigma.
|
+ watched (Harper) lecture 15 on Sets being closed under type formations ->- motivates showing in Agda Sets closed under Sigma.
|
||||||
- Harper does product case, claiming sigma case follows analogously,
|
+ Harper does product case, claiming sigma case follows analogously,
|
||||||
- attempt proof in Cubical Agda but highly non-obvious how to use that fibers are Sets.
|
+ attempt proof in Cubical Agda but highly non-obvious how to use that fibers are Sets.
|
||||||
- difficulty is that PathP not in one fiber, but PathOver is, AND PathOver <-> PathP NON-obvious
|
+ difficulty is that PathP not in one fiber, but PathOver is, AND PathOver <-> PathP NON-obvious
|
||||||
- Easy to generalize situation to n-types being closed under Sigma (7.1.8 in HoTT book), we showed this assuming PathPIsoPath
|
+ Easy to generalize situation to n-types being closed under Sigma (7.1.8 in HoTT book), we showed this assuming PathPIsoPath
|
||||||
|
|||||||
56
README.md
Normal file
56
README.md
Normal file
@ -0,0 +1,56 @@
|
|||||||
|
The HoTT Game
|
||||||
|
=============
|
||||||
|
|
||||||
|
The Homotopy Type Theory (HoTT) Game is a project aimed at
|
||||||
|
introducing mathematicians with no experience
|
||||||
|
in proof verification interested in HoTT and able to use Agda for HoTT.
|
||||||
|
This guide will help you get the Game working for you.
|
||||||
|
|
||||||
|
## Installing Agda and the Cubical Agda library
|
||||||
|
|
||||||
|
Our Game is in Agda, which can be installed following instructions
|
||||||
|
[on their site](
|
||||||
|
https://agda.readthedocs.io/en/latest/getting-started/installation.html).
|
||||||
|
It is recommended that you use Agda in the text editor
|
||||||
|
[emacs](
|
||||||
|
https://www.gnu.org/software/emacs/tour/index.html),
|
||||||
|
in particular
|
||||||
|
[Doom Emacs](https://github.com/hlissner/doom-emacs) is a bit nicer if you
|
||||||
|
can't be bothered to do a bunch of configuration.
|
||||||
|
|
||||||
|
Once you have Emacs and Agda, get the [Cubical Library](
|
||||||
|
https://github.com/agda/cubical) (version 0.3)
|
||||||
|
and make sure Agda knows where your cubical library is
|
||||||
|
by following instructions on the [library management page](
|
||||||
|
https://agda.readthedocs.io/en/latest/tools/package-system.html?highlight=library%20management).
|
||||||
|
(In short: locate (or create) your `libraries` file and add a line
|
||||||
|
```
|
||||||
|
the-directory/cubical.agda-lib
|
||||||
|
```
|
||||||
|
to it, where `the-directory` is the location of `cubical.agda-lib` on your computer.)
|
||||||
|
|
||||||
|
Get the HoTT Game by [cloning this repository](
|
||||||
|
https://git-scm.com/book/en/v2/Git-Basics-Getting-a-Git-Repository)
|
||||||
|
into a folder and then making sure that Agda knows where the HoTT Game is
|
||||||
|
by adding a line
|
||||||
|
```
|
||||||
|
the-directory/HoTTGameLib.agda
|
||||||
|
```
|
||||||
|
to your `libraries` file as above.
|
||||||
|
|
||||||
|
Try opening up `Trinitarianism/AsProps/Quest0.agda` in Emacs
|
||||||
|
and do `Ctrl-c Ctrl-l`.
|
||||||
|
Some text should be highlighted, and any `?` should turn into `{ }`.
|
||||||
|
|
||||||
|
## How the game works
|
||||||
|
|
||||||
|
Our Game is currently under development. Please contact the devs.
|
||||||
|
Currently you can try `Trinitarianism/AsProps/Quest0.agda`,
|
||||||
|
making use of the accompanying solutions Agda file.
|
||||||
|
|
||||||
|
## Emacs and Agda usage
|
||||||
|
We have a file with a list of basic Emacs commands and
|
||||||
|
you _should_ be able to learn how to use Agda as you go along.
|
||||||
|
|
||||||
|
## Feedback
|
||||||
|
If you have any feedback please contact the devs.
|
||||||
@ -1,4 +1,4 @@
|
|||||||
module TheHoTTGame.Trinitarianism.AsCats where
|
module Trinitarianism.AsCats where
|
||||||
|
|
||||||
{-
|
{-
|
||||||
Here are some things that we could like to have in a category
|
Here are some things that we could like to have in a category
|
||||||
|
|||||||
@ -1,5 +1,5 @@
|
|||||||
module TheHoTTGame.Trinitarianism.AsProps.Quest0 where
|
module Trinitarianism.AsProps.Quest0 where
|
||||||
open import TheHoTTGame.Trinitarianism.AsProps.Quest0Preamble
|
open import Trinitarianism.AsProps.Quest0Preamble
|
||||||
|
|
||||||
{-
|
{-
|
||||||
Here are some things that we could like to have in a logical framework
|
Here are some things that we could like to have in a logical framework
|
||||||
|
|||||||
@ -1,5 +1,5 @@
|
|||||||
|
|
||||||
module TheHoTTGame.Trinitarianism.AsProps.Quest0Preamble where
|
module Trinitarianism.AsProps.Quest0Preamble where
|
||||||
|
|
||||||
open import Cubical.Core.Everything hiding (_∨_) public
|
open import Cubical.Core.Everything hiding (_∨_) public
|
||||||
open import Cubical.Data.Nat public
|
open import Cubical.Data.Nat public
|
||||||
|
|||||||
@ -1,5 +1,5 @@
|
|||||||
module TheHoTTGame.Trinitarianism.AsProps.Quest0Solutions where
|
module Trinitarianism.AsProps.Quest0Solutions where
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open import TheHoTTGame.Trinitarianism.AsProps.Quest0Preamble
|
open import Trinitarianism.AsProps.Quest0Preamble
|
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|
|
||||||
data ⊤ : Prop where
|
data ⊤ : Prop where
|
||||||
trivial : ⊤
|
trivial : ⊤
|
||||||
|
|||||||
@ -0,0 +1,13 @@
|
|||||||
|
{-
|
||||||
|
Two things being equal is also a proposition
|
||||||
|
|
||||||
|
-}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
-- FalseToTrue : ⊥ → ⊤
|
||||||
|
-- FalseToTrue = λ x → trivial
|
||||||
|
|
||||||
|
-- FalseToTrue' : ⊥ → ⊤
|
||||||
|
-- FalseToTrue' ()
|
||||||
@ -1,31 +0,0 @@
|
|||||||
{- This says that ⊥ is the proposition where there are no proofs of it. -}
|
|
||||||
|
|
||||||
{-
|
|
||||||
Given two propositions P and Q, we can form a new proposition 'P implies Q'
|
|
||||||
written P → Q
|
|
||||||
To introduce a proof of P → Q we assume a proof x of P and give a proof y of Q
|
|
||||||
|
|
||||||
Here is an example demonstrating → in action
|
|
||||||
-}
|
|
||||||
-- TrueToTrue : ⊤ → ⊤
|
|
||||||
-- TrueToTrue = ?
|
|
||||||
|
|
||||||
{-
|
|
||||||
* press C-c C-l (this means Ctrl-c Ctrl-l) to load the document,
|
|
||||||
and now you can fill the holes
|
|
||||||
* press C-c C-r and agda will try to help you,
|
|
||||||
* you should see λ x → { }
|
|
||||||
* navigate to the hole { } using C-c C-f (forward) or C-c C-b (backward)
|
|
||||||
* to check what agda wants in the hole use C-c C-,
|
|
||||||
* the Goal area should look like
|
|
||||||
Goal: ⊤
|
|
||||||
————————————————————————————————————————————————————————————
|
|
||||||
x : ⊤
|
|
||||||
|
|
||||||
* this means you have a proof x : ⊤ and you need to give a proof of ⊤
|
|
||||||
* you can now give it a proof of ⊤ and press C-c C-SPC to fill the hole
|
|
||||||
|
|
||||||
There is more than one proof (see answers) - are they the same?
|
|
||||||
-}
|
|
||||||
|
|
||||||
-- solutions:
|
|
||||||
@ -1,13 +0,0 @@
|
|||||||
{-
|
|
||||||
Two things being equal is also a proposition
|
|
||||||
|
|
||||||
-}
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
-- FalseToTrue : ⊥ → ⊤
|
|
||||||
-- FalseToTrue = λ x → trivial
|
|
||||||
|
|
||||||
-- FalseToTrue' : ⊥ → ⊤
|
|
||||||
-- FalseToTrue' ()
|
|
||||||
BIN
_build/2.6.2/agda/Trinitarianism/AsCats.agdai
Normal file
BIN
_build/2.6.2/agda/Trinitarianism/AsCats.agdai
Normal file
Binary file not shown.
BIN
_build/2.6.2/agda/Trinitarianism/AsProps/Quest0Preamble.agdai
Normal file
BIN
_build/2.6.2/agda/Trinitarianism/AsProps/Quest0Preamble.agdai
Normal file
Binary file not shown.
BIN
_build/2.6.2/agda/Trinitarianism/AsProps/Quest0Solutions.agdai
Normal file
BIN
_build/2.6.2/agda/Trinitarianism/AsProps/Quest0Solutions.agdai
Normal file
Binary file not shown.
Loading…
Reference in New Issue
Block a user