From 064691b0291896bb10fffcef637ff085be7606bb Mon Sep 17 00:00:00 2001 From: kl-i Date: Thu, 29 Jul 2021 22:55:16 +0100 Subject: [PATCH] Checked Trinitarianism. --- README.md | 34 +++++--- Trinitarianism/.DS_Store | Bin 0 -> 6148 bytes Trinitarianism/Quest0.md | 49 +++++++----- Trinitarianism/Quest1.agda | 6 ++ Trinitarianism/Quest1.md | 74 ++++++++++-------- Trinitarianism/Quest2.md | 37 +++++---- Trinitarianism/README.md | 26 +++--- .../agda/Trinitarianism/Preambles/P1.agdai | Bin 0 -> 27266 bytes .../agda/Trinitarianism/Preambles/P2.agdai | Bin 0 -> 23534 bytes .../agda/Trinitarianism/Quest1Solutions.agdai | Bin 0 -> 29187 bytes .../agda/Trinitarianism/Quest2Solutions.agdai | Bin 0 -> 28993 bytes 11 files changed, 134 insertions(+), 92 deletions(-) create mode 100644 Trinitarianism/.DS_Store create mode 100644 _build/2.6.3/agda/Trinitarianism/Preambles/P1.agdai create mode 100644 _build/2.6.3/agda/Trinitarianism/Preambles/P2.agdai create mode 100644 _build/2.6.3/agda/Trinitarianism/Quest1Solutions.agdai create mode 100644 _build/2.6.3/agda/Trinitarianism/Quest2Solutions.agdai diff --git a/README.md b/README.md index 9b097e7..8465b57 100644 --- a/README.md +++ b/README.md @@ -1,10 +1,11 @@ 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. +The Homotopy Type Theory (HoTT) Game is a project written by mathematicians +for mathematicians interested in HoTT and no experience in proof verification, +with the aim of introducing Cubical Agda as a tool for +trying out mathematics in HoTT. +This page will help you get the Game working for you. ## Installing Agda and the Cubical Agda library @@ -15,8 +16,8 @@ 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. +[Doom Emacs](https://github.com/hlissner/doom-emacs), +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) @@ -38,19 +39,28 @@ 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`. +Try opening `Trinitarianism/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 under development. Please contact the devs. -Currently you can try `Trinitarianism/AsProps/Quest0.agda`, -making use of the accompanying solutions Agda file. +Currently you can try the _quests_ in the `Trinitarianism` folder. +Each quest consists of three files, for example : +- `Trinitarianism/Quest0.md` is the guide for the quest + In there, you will find details of the tasks + you must finish in order to complete the quest. + For now, it is recommended that + you view these online within github. +- `Trinitarianism/Quest0.agda` is the actual file in which + you do the quest. +- `Trinitarianism/Quest0Solutions.agda` contains + solutions to the tasks in the quest. ## 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. +We will have a file with a list of basic Emacs commands, +but you _should_ be able to learn how to use Agda as you go along. ## Feedback If you have any feedback please contact the devs. diff --git a/Trinitarianism/.DS_Store b/Trinitarianism/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..5008ddfcf53c02e82d7eee2e57c38e5672ef89f6 GIT binary patch literal 6148 zcmeH~Jr2S!425mzP>H1@V-^m;4Wg<&0T*E43hX&L&p$$qDprKhvt+--jT7}7np#A3 zem<@ulZcFPQ@L2!n>{z**++&mCkOWA81W14cNZlEfg7;MkzE(HCqgga^y>{tEnwC%0;vJ&^%eQ zLs35+`xjp>T0 -This has no interpretation as a proposition since +`ℕ` has no interpretation as a proposition since there are 'too many proofs' - mathematicians classically don't distinguish between proofs of a single proposition. @@ -151,16 +157,19 @@ go to Quest1! You may have noticed the notational similarities between `zero : ℕ` and `ℕ : Type`. This may have lead you to the question, `Type : ?`. -In type theory, we simply assert `Type : Type 1`. -But then we are chasing our tail, asking `Type 1 : ?`. -Type theorists make sure that every type (the thing on the right side of the `:`) -itself is a term, and every term has a type. +In type theory, we simply assert `Type : Type₁`. +But then we are chasing our tail, asking `Type₁ : Type₂`. +Type theorists make sure that every type +(i.e. anything the right side of `:`) +itself is a term (i.e. anything on the left of `:`), +and every term has a type. So what we really need is ``` Type : Type₁, Type₁ : Type₂, Type₂ : Type₃, ⋯ ``` These are called _universes_. The numberings of universes are called _levels_. +We will start using universes in the next quest. diff --git a/Trinitarianism/Quest2.md b/Trinitarianism/Quest2.md index 728d49a..2ac137c 100644 --- a/Trinitarianism/Quest2.md +++ b/Trinitarianism/Quest2.md @@ -3,8 +3,8 @@ We will try to formulate and prove the statement > The sum of two even naturals is even. To do so we must define `+` on the naturals. -Addition takes in two naturals and spits out a -natural, so it should have type `ℕ → ℕ → ℕ`. +Addition takes in two naturals and spits out a natural, +so it should have type `ℕ → ℕ → ℕ`. ```agda _+_ : ℕ → ℕ → ℕ n + m = ? @@ -12,8 +12,8 @@ n + m = ? Agda supports the notation `_+_` (without spaces) which means from now on you can write `0 + 1` and so on (with spaces). -Try coming up with a sensible definition yourself, -it may not look exactly like ours. +Try coming up with a sensible definition. +It may not look 'the same' as ours.
Hint `n + 0` should be `n` and `n + (m + 1)` should be `(n + m) + 1` @@ -38,27 +38,30 @@ There are three ways to interpret this: - `isEven (_ .fst + _ .fst)` is a bundle over the categorical product `Σ ℕ isEven × Σ ℕ isEven` and `SumOfEven` is a _section_ of the bundle. -More generally given `A : Type` and `B : A → Type` we can form the _pi type_ -`(x : A) → B x : Type` (in other languages `Π (x : ℕ), isEven n`). +More generally given `A : Type` and `B : A → Type` +we can form the _pi type_ `(x : A) → B x : Type` +(in other languages `Π (x : ℕ), isEven n`). The notation suggests that these behave like functions, and indeed in the special case where the fiber is constant -with respect to the base a section is just a term of type `A → B`, -i.e. a function. Hence pi types are also known as -_dependent function types_. +with respect to the base space +a section is just a term of type `A → B`, i.e. a function. +Hence pi types are also known as _dependent function types_. We are now in a position to prove the statement. Have fun! _Important_: Once you have proven the statement, check out our two ways of defining addition `_+_` and `_+'_` (in the solutions). -Use `C-c C-n` to check that they compute the same values -on different examples. -Uncomment the code for `Sum'OfEven` in the solutions, -it is just `SumOfEven` but with `+`s changed for `+'`s. -Load the file. Does the proof still work? +- Use `C-c C-n` to check that they compute the same values + on different examples. +- Uncomment the code for `Sum'OfEven` in the solutions. + It is just `SumOfEven` but with `+`s changed for `+'`s. +- Load the file. Does the proof still work? -In our proof of `SumOfEven` we explicitely used the definition of `_+_`, -which means that if we wanted to use our proof on someone else's -definition of addition, things might break. +Our proof `SumOfEven` relied on +the explicit definition of `_+_`, +which means if we wanted to use our proof on +someone else's definition of addition, +it might not work anymore. > But `_+_` and `_+'_` compute the same values. > Are `_+_` and `_+'_` 'the same'? What is 'the same'? diff --git a/Trinitarianism/README.md b/Trinitarianism/README.md index 9d5dc9a..85e12c8 100644 --- a/Trinitarianism/README.md +++ b/Trinitarianism/README.md @@ -1,7 +1,7 @@ Trinitarianism ============== -By the end of this arc we will have 'a place to do maths'. +By the end of this arc we will (almost) have 'a place to do maths'. The 'types' that will populated this 'place' will have three interpretations: - Proof theoretically, with types as propositions @@ -16,12 +16,14 @@ will have three interpretations: ## Terms and Types Here are some things that we could like to have in a 'place to do maths' - - objects to reason about (like ℕ) - - recipes for making things inside objects (like + 1) - - propositions to reason with (with the data of proofs) (like _ = 0) + - objects to reason about (E.g. `ℕ`) + - recipes for making things inside objects + (E.g. `n + m` for `n` and `m` in naturals.) + - propositions to reason with (E.g. `n = 0` for `n` in naturals.) In proof theory, types are propositions and terms of a type are their proofs. -In type theory, types are programs and terms are algorithms. +In type theory, types are programs / constructions and +terms are algorithms / recipes. In category theory, types are objects and terms are generalised elements. ## Non-dependent Types @@ -34,10 +36,14 @@ In category theory, types are objects and terms are generalised elements. ## Dependent Types -- predicate / type family / over category -- substitution / substitution / pullback -- existence / Σ type / left adjoint to pullback -- for all / Π type / right adjoint to pullback +- predicate / type family / bundle +- substitution / substitution / pullback (of bundles) +- existence / Σ type / total space of bundles +- for all / Π type / space of sections of bundles +## Something doesn't feel the Same -> Question: how do we talk about equality? +There will be one thing missing from this 'place to do maths' +and that is a notion of _equality_. +This is where HoTT deviates from its predecessors, +and is the theme of the next arc. diff --git a/_build/2.6.3/agda/Trinitarianism/Preambles/P1.agdai b/_build/2.6.3/agda/Trinitarianism/Preambles/P1.agdai new file mode 100644 index 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