From 6c9f228adebf43629e0b2f68636a1dd3a585c844 Mon Sep 17 00:00:00 2001 From: jlh Date: Wed, 21 Jul 2021 17:14:03 +0100 Subject: [PATCH] trinit --- Trinitarianism/Quest0.agda | 176 ++++++++++++++---- Trinitarianism/Quest0.md | 144 ++++++++++++++ Trinitarianism/Quest0Preamble.agda | 5 - Trinitarianism/images/nno.png | Bin 0 -> 73057 bytes .../agda/Trinitarianism/Quest0Preamble.agdai | Bin 22919 -> 11329 bytes 5 files changed, 282 insertions(+), 43 deletions(-) create mode 100644 Trinitarianism/Quest0.md create mode 100644 Trinitarianism/images/nno.png diff --git a/Trinitarianism/Quest0.agda b/Trinitarianism/Quest0.agda index cb64a87..9aeccb3 100644 --- a/Trinitarianism/Quest0.agda +++ b/Trinitarianism/Quest0.agda @@ -4,68 +4,165 @@ open import Trinitarianism.Quest0Preamble private postulate u : Level + A : Type u + {- - 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) +There are three ways of looking at `A : Type u`. + - proof theoretically, '`A` is a proposition' + - type theoretically, '`A` is a construction' + - categorically, '`A` is an object in category `Type u`' - To make propositions we want - * False ⊥ - * True ⊤ - * Or ∨ - * And ∧ - * Implication → +We will explain what u : Level and Type u is at the end. - but propositions are useless if they're not talking about anything, - so we also want - * Predicates - * Exists ∃ - * For all ∀ - * Equality ≡ (of objects) +A first example of a type construction is the function type. +Given types `A` and `B`, we have another type `A → B` which can be seen as + - the proposition '`A` implies `B`' + - the construction 'ways to convert `A` recipes to `B` recipes' + - internal hom of the category `Type u` + +To give examples of this, let's make some types first! -} -- Here is how we define 'true' data ⊤ : Type u where trivial : ⊤ -{- It reads '⊤ is a proposition and there is a proof of it, called "trivial"'. -} - --- Here is how we define 'false' -data ⊥ : Type u where - -{- 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 +It reads '`⊤` is an inductive type with a constructor `trivial`', +with three interpretations + - `⊤` is a proposition and there is a proof of it, called `trivial`. + - `⊤` is a construction with a recipe called `trivial` + - `⊤` is a terminal object: every object has a morphism into `⊤` given by `· ↦ trivial` + +The above tells you how we _make_ a term of type `⊤`, +let's see an example of _using_ a term of type `⊤`: -} TrueToTrue : ⊤ → ⊤ TrueToTrue = {!!} {- - * press C-c C-l (this means Ctrl-c Ctrl-l) to load the document, + - press `C-c C-l` (this means `Ctrl-c Ctrl-l`) to load the document, and now you can fill the holes - * navigate to the hole { } using C-c C-f (forward) or C-c C-b (backward) - * press C-c C-r and agda will try to help you (r for refine) - * you should see λ x → { } - * navigate to the new hole - * C-c C-, to check what agda wants in the hole (C-c C-comma) - * the Goal area should look like + - navigate to the hole `{ }` using `C-c C-f` (forward) or `C-c C-b` (backward) + - press `C-c C-r` and agda will try to help you (r for refine) + - you should see `λ x → { }` + - navigate to the new hole + - `C-c C-,` to check the goal (`C-c C-comma`) + - the Goal area should look like + ``` Goal: ⊤ - ———————————————————————————————————————————————————————————— + ————————————————————————— x : ⊤ + ``` - * this means you have a proof of ⊤ '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 + - you have a proof/recipe/generalized element `x : ⊤` + and you need to give a p/r/g.e. of `⊤` + - you can give it a p/r/g.e. of `⊤` and press `C-c C-SPC` to fill the hole - There is more than one proof (see solutions) - are they the same? +There is more than one proof (see solutions) - are they the same? +Here is an important one: -} +TrueToTrue' : ⊤ → ⊤ +TrueToTrue' x = {!!} + +{- + + - Naviagate to the hole and check the goal. + - Note `x` is already taken out for you. + - You can try type `x` in the hole and `C-c C-c` + - `c` stands for 'cases on `x`' and the only case is `trivial`. + +Built into the definition of `⊤` is agda's way of making a map out of ⊤ +into another type A, which we have just used. +It says to map out of ⊤ it suffices to do the case when `x` is `trivial`, or + - the only proof of `⊤` is `trivial` + - the only recipe for `⊤` is `trivial` + - the only one generalized element `trivial` in `⊤` +-} + + + +-- Here is how we define 'false' +data ⊥ : Type u where + +{- + +It reads '`⊥` is an inductive type with no constructors', +with three interepretations + - `⊥` is a proposition with no proofs + - `⊥` is a construction with no recipes + - There are no generalized elements of `⊥` (it is a strict initial object) + +Let's try mapping out of `⊥`. +-} + +explosion : ⊥ → ⊤ +explosion x = {!!} + +{- + + - Navigate to the hole and do cases on `x`. + +Agda knows that there are no cases so there is nothing to do! +This has three interpretations: + - false implies anything (principle of explosion) + - ? + - ⊥ is initial in the category `Type u` + +-} + +{- +We can also encode "natural numbers" as a type. +-} +data ℕ : Type where + zero : ℕ + suc : ℕ → ℕ +{- +As a construction, this reads ' + - ℕ is a type of construction + - `zero` is a recipe for ℕ + - `suc` takes an existing recipe for ℕ and gives + another recipe for ℕ. + ' + +We can see `ℕ` as a categorical notion: +ℕ is a natural numbers object in the category `Type u`, +with `zero : ⊤ → ℕ` and `suc : ℕ → ℕ` such that +given any `⊤ → A → A` there exist a unique morphism `ℕ → A` +such that the diagram commutes: + + +This has no interpretation as a proposition since +there are too many terms, +since mathematicians classically didn't distinguish +between proofs of the same thing. +(ZFC doesn't even mention logic internally, +unlike Type Theory!) + + + + +-} + + + + + + + + + + + + + + + + + {- Let's assume we have the following the naturals ℕ @@ -218,3 +315,6 @@ You need to first uncomment this by getting rid of the -- in front (C-x C-;) -- DecidableIsZero : (n : ℕ) → {!!} -- DecidableIsZero zero = {!!} -- DecidableIsZero (suc n) = {!!} + +-- TODO terms and types +-- TODO universe levels diff --git a/Trinitarianism/Quest0.md b/Trinitarianism/Quest0.md new file mode 100644 index 0000000..eb6f4b4 --- /dev/null +++ b/Trinitarianism/Quest0.md @@ -0,0 +1,144 @@ +There are three ways of looking at `A : Type u`. + - proof theoretically, '`A` is a proposition' + - type theoretically, '`A` is a construction' + - categorically, '`A` is an object in category `Type u`' + +We will explain what u : Level and Type u is at the end. + +A first example of a type construction is the function type. +Given types `A` and `B`, we have another type `A → B` which can be seen as + - the proposition '`A` implies `B`' + - the construction 'ways to convert `A` recipes to `B` recipes' + - internal hom of the category `Type u` + +To give examples of this, let's make some types first! + +```agda + +-- Here is how we define 'true' +data ⊤ : Type u where + trivial : ⊤ + +``` + +It reads '`⊤` is an inductive type with a constructor `trivial`', +with three interpretations + - `⊤` is a proposition and there is a proof of it, called `trivial`. + - `⊤` is a construction with a recipe called `trivial` + - `⊤` is a terminal object: every object has a morphism into `⊤` given by `· ↦ trivial` + +The above tells you how we _make_ a term of type `⊤`, +let's see an example of _using_ a term of type `⊤`: + +```agda +TrueToTrue : ⊤ → ⊤ +TrueToTrue = {!!} +``` + + - press `C-c C-l` (this means `Ctrl-c Ctrl-l`) to load the document, + and now you can fill the holes + - navigate to the hole `{ }` using `C-c C-f` (forward) or `C-c C-b` (backward) + - press `C-c C-r` and agda will try to help you (r for refine) + - you should see `λ x → { }` + - navigate to the new hole + - `C-c C-,` to check the goal (`C-c C-comma`) + - the Goal area should look like + + ``` + Goal: ⊤ + ————————————————————————— + x : ⊤ + ``` + + - you have a proof/recipe/generalized element `x : ⊤` + and you need to give a p/r/g.e. of `⊤` + - you can give it a p/r/g.e. of `⊤` and press `C-c C-SPC` to fill the hole + +There is more than one proof (see solutions) - are they the same? +Here is an important one: + +```agda + +TrueToTrue' : ⊤ → ⊤ +TrueToTrue' x = {!!} + +``` + + + - Naviagate to the hole and check the goal. + - Note `x` is already taken out for you. + - You can try type `x` in the hole and `C-c C-c` + - `c` stands for 'cases on `x`' and the only case is `trivial`. + +Built into the definition of `⊤` is agda's way of making a map out of ⊤ +into another type A, which we have just used. +It says to map out of ⊤ it suffices to do the case when `x` is `trivial`, or + - the only proof of `⊤` is `trivial` + - the only recipe for `⊤` is `trivial` + - the only one generalized element `trivial` in `⊤` + +```agda + +-- Here is how we define 'false' +data ⊥ : Type u where + +``` + + +It reads '`⊥` is an inductive type with no constructors', +with three interepretations + - `⊥` is a proposition with no proofs + - `⊥` is a construction with no recipes + - There are no generalized elements of `⊥` (it is a strict initial object) + +Let's try mapping out of `⊥`. + +```agda +explosion : ⊥ → ⊤ +explosion x = {!!} +``` + + - Navigate to the hole and do cases on `x`. + +Agda knows that there are no cases so there is nothing to do! +This has three interpretations: + - false implies anything (principle of explosion) + - ? + - `⊥` is initial in the category `Type u` + + +We can also encode "natural numbers" as a type. + +```agda +data ℕ : Type where + zero : ℕ + suc : ℕ → ℕ +``` + +As a construction, this reads ' + - `ℕ` is a type of construction + - `zero` is a recipe for `ℕ` + - `suc` takes an existing recipe for `ℕ` and gives + another recipe for `ℕ`. + ' + +We can see `ℕ` as a categorical notion: +ℕ is a natural numbers object in the category `Type u`, +with `zero : ⊤ → ℕ` and `suc : ℕ → ℕ` such that +given any `⊤ → A → A` there exist a unique morphism `ℕ → A` +such that the diagram commutes: +![nno](images/nno.png) + + +This has no interpretation as a proposition since +there are too many terms, +since mathematicians classically didn't distinguish +between proofs of the same thing. +(ZFC doesn't even mention logic internally, +unlike Type Theory!) + + + + +-} + diff --git a/Trinitarianism/Quest0Preamble.agda b/Trinitarianism/Quest0Preamble.agda index aa45513..d5564dd 100644 --- a/Trinitarianism/Quest0Preamble.agda +++ b/Trinitarianism/Quest0Preamble.agda @@ -2,8 +2,3 @@ module Trinitarianism.Quest0Preamble where open import Cubical.Core.Everything hiding (_∨_) public -open import Cubical.Data.Nat public - -private - postulate - u : Level diff --git a/Trinitarianism/images/nno.png b/Trinitarianism/images/nno.png new file mode 100644 index 0000000000000000000000000000000000000000..0543959dbd896475aa1f04b55b674720f35c8381 GIT binary patch literal 73057 zcmeFZbySq?+CEH&NU4BQA}K8`LrW`-(%p@~&@GB6-GU4q($dl>-ObR0L3d04uEFQo z@7}*}zw7(=`__6`3m9hRp8LMe>x|<#&hzG-ijpk;4e}c(C@A>ya#Ct2D45A8DCi?N zSm2qxXO7({C^yZmB_&nlB_(N8oEqgBFm8Yqm#=67pTe zK`2Wu*M!TAP7q48XGgMZ<5By*g0DfpI5ws0&FoE-q>klY7t|s9x6T@6G-T7mzBp6* zEOBzELp-l2*@*VvM8!UcWMNCm{c#cnOK-Zl$I-R1F=)4R+3$Ds*;iD|{R5vXb89Y^T#2oCi_Z2!XZ(|JsZZTPC=Xv0 zg((fMWG$g#k}VNFdQgvd@|`QH|AFjNydNf2dmEJU^n~1c>{joxLlTRq>gBRSDBcC3H`?B%KQYU>HPb6Up{?Xpke#;;wp)jWVRnLRw9!CyFZC#AHK-?TM<#Yeu&sNqWWMj?lSyk~a)Cok!g_ZdIN>)_zWu 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