Trinit/Quest2 added

Plan.org

@@ -6,7 +6,7 @@
 
 ** 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
+  - Big-ass-boss: Loop space of S^1 = Z
   - Try to balance hiding cubical implementations whilst exploiting their advantages
 
 ** Barriers
@@ -76,3 +76,13 @@
   + 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
+
+
+**  Minibosses
+- _+_ unique on the naturals
+  + axiomatize addition on naturals
+  + naturals is a set
+  + fun extensionality
+  + contractability
+  + propositions
+  + propositions closed under sigma types

Trinitarianism/Preambles/P0.agda

@@ -1,3 +1,3 @@
-module Trinitarianism.Quest0Preamble where
+module Trinitarianism.Preambles.P0 where
 
 open import Cubical.Core.Everything hiding (_∨_) public

Trinitarianism/Preambles/P1.agda

@@ -0,0 +1,6 @@
+module Trinitarianism.Preambles.P1 where
+
+open import Cubical.Core.Everything public
+open import Cubical.Data.Unit public renaming (Unit to ⊤)
+open import Cubical.Data.Empty public using (⊥)
+open import Cubical.Data.Nat public hiding (isEven)

Trinitarianism/Preambles/P2.agda

@@ -0,0 +1,5 @@
+module Trinitarianism.Preambles.P2 where
+
+open import Cubical.Core.Everything public
+open import Cubical.Data.Nat public hiding (_+_ ; isEven)
+open import Trinitarianism.Quest1Solutions public

Trinitarianism/Quest0.agda

@@ -1,5 +1,5 @@
 module Trinitarianism.Quest0 where
-open import Trinitarianism.Quest0Preamble
+open import Trinitarianism.Preambles.P0
 
 data ⊤ : Type where
   tt : ⊤

Trinitarianism/Quest0.md

@@ -59,8 +59,10 @@ TrueToTrue = {!!}
   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:
+There is more than one proof (see solutions) - are they 'the same'?
+What is 'the same'?
+
+Here is an important solution:
 
 ```agda
 TrueToTrue' : ⊤ → ⊤

Trinitarianism/Quest0Solutions.agda

@@ -1,5 +1,5 @@
 module Trinitarianism.Quest0Solutions where
-open import Trinitarianism.Quest0Preamble
+open import Trinitarianism.Preambles.P0
 
 
 data ⊤ : Type where

Trinitarianism/Quest1.agda

@@ -1,7 +1,6 @@
 module Trinitarianism.Quest1 where
 
-open import Cubical.Core.Everything
-open import Cubical.Data.Nat hiding (isEven)
+open import Trinitarianism.Preambles.P1
 
 isEven : ℕ → Type
 isEven n = {!!}

Trinitarianism/Quest1.md

@@ -1,4 +1,4 @@
-# Dependent Types
+# Dependent Types and Sigma Types
 
 In a 'place to do maths'
 we would like to be able to express and 'prove'
@@ -84,6 +84,7 @@ There are three interpretations of `isEven : ℕ → Type`.
   i.e. an object in the over-category `Type↓ℕ`.
   Pictorially, it looks like
 
+
   <img src="images/isEven.png" 
      alt="isEven" 
      width="500"/>
@@ -92,6 +93,9 @@ There are three interpretations of `isEven : ℕ → Type`.
   `isEven n` is called the _fiber over `n`_.
   In this particular example the fibers are either empty
   or singleton.
+
+In general given a type `A : Type`, 
+a _dependent type over `A`_ is a term of type `A → Type`.
   
 You can check if `2` is even by asking agda to 'normalize' it:
 do `C-c C-n` (`n` for normalize) and type in `isEven 2`.
@@ -140,6 +144,17 @@ where `a : A` and `b : B a`.
 
 There are two ways of using a term in a sigma type.
 We can extract the first part using `fst` or the second part using `snd`.
+Given `x : Σ A B` there are three interpretations of `fst` and `snd`:
+- Viewing `x` as a proof of existence
+  `fst x` provides the witness of existence and `snd` provides the proof 
+  of the property
+- Viewing `x` as a recipe `fst` extracts the first component and 
+  `snd` extracts the second component 
+- Viewing `x` as a point in the total space of a bundle 
+  `fst x` is the point that `x` is over in the base space and `snd x`
+  is the point in the fiber that `x` represents.
+  In particular you can interpret `fst` as projection from the total space
+  to the base space, collapsing fibers.
 For example to define a map that takes an even natural and divides it by two 
 we can 
 ```agda
@@ -147,7 +162,7 @@ div2 : Σ ℕ isEven → ℕ
 div2 x = ? 
 ```
 - Load the file, go to the hole and case on `x` 
-  (you might want to rename `fst₁` and `snd₁`).
+  (you might want to rename `fst₁` and `snd₁`). 
 ```agda
 div2 : Σ ℕ isEven → ℕ
 div2 (fst₁ , snd₁) = {!!}
@@ -174,3 +189,4 @@ since if they were `div2 (2 , tt)` would be 'the same' `div2 (36/2 , tt)`,
 hence `1` would be 'the same' as `18`. 
 > Are they 'the same'? What is 'the same'? 
 <!-- see Arc/Quest smth? -->
+

Trinitarianism/Quest1Solutions.agda

@@ -1,9 +1,6 @@
 module Trinitarianism.Quest1Solutions where
 
-open import Cubical.Core.Everything
-open import Cubical.Data.Unit renaming (Unit to ⊤)
-open import Cubical.Data.Empty
-open import Cubical.Data.Nat hiding (isEven)
+open import Trinitarianism.Preambles.P1
 
 isEven : ℕ → Type
 isEven zero = ⊤

Trinitarianism/Quest2.agda

@@ -0,0 +1,9 @@
+module Trinitarianism.Quest2 where
+
+open import Trinitarianism.Preambles.P2
+
+_+_ : ℕ → ℕ → ℕ
+n + m = {!!}
+
+SumOfEven : (x : Σ ℕ isEven) → (y : Σ ℕ isEven) → isEven (x .fst + y .fst)
+SumOfEven x y = {!!}

Trinitarianism/Quest2.md

@@ -0,0 +1,64 @@
+# Pi Types
+
+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`.
+```agda
+_+_ : ℕ → ℕ → ℕ
+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.
+<details>
+  <summary>Hint</summary>
+  `n + 0` should be `n` and `n + (m + 1)` should be `(n + m) + 1`
+</details>
+
+Now we can make the statement:
+```agda
+SumOfEven : (x : Σ ℕ isEven) → (y : Σ ℕ isEven) → isEven (x .fst + y .fst)
+SumOfEven x y = ?
+```
+> Tip: `x .fst` is another notation for `fst x`.
+> This works for all sigma types.
+There are three ways to interpret this:
+- For all even naturals `x` and for all even naturals `y`, 
+  their sum is even.
+- `isEven (x .fst + y .fst)` is a construction depending on two recipes
+  `x` and `y`.
+  Given two recipes `x` and `y` of `Σ ℕ isEven`, 
+  we break them down into their first components,
+  apply the conversion `_+_`,
+  and form a recipe for `isEven` of the result.
+- `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`). 
+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_.
+
+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?
+
+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.
+> But `_+_` and `_+'_` compute the same values. 
+> Are `_+_` and `_+'_` 'the same'? What is 'the same'?

Trinitarianism/Quest2Solutions.agda

@@ -0,0 +1,23 @@
+module Trinitarianism.Quest2Solutions where
+
+open import Trinitarianism.Preambles.P2
+
+_+_ : ℕ → ℕ → ℕ
+n + zero = n
+n + suc m = suc (n + m)
+
+_+'_ : ℕ → ℕ → ℕ
+zero +' n = n
+suc m +' n = suc (m +' n)
+
+SumOfEven : (x : Σ ℕ isEven) → (y : Σ ℕ isEven) → isEven (x .fst + y .fst)
+SumOfEven x (zero , hy) = x .snd
+SumOfEven x (suc (suc y) , hy) = SumOfEven x (y , hy)
+
+{-
+
+Sum'OfEven : (x : Σ ℕ isEven) → (y : Σ ℕ isEven) → isEven (x .fst +' y .fst)
+Sum'OfEven x (zero , hy) = x .snd
+Sum'OfEven x (suc (suc y) , hy) = Sum'OfEven x (y , hy)
+
+-}