From 35f0774d46ef70499866088b4b449cf4f75c91f6 Mon Sep 17 00:00:00 2001
From: Jlh18 <josephhua73@yahoo.com>
Date: Mon, 16 Aug 2021 19:08:50 +0100
Subject: [PATCH] moved furniature

---
 0Trinitarianism/Quest0.md | 12 +++++++++++-
 0Trinitarianism/Quest1.md |  5 ++++-
 0Trinitarianism/Quest2.md |  4 +++-
 3 files changed, 18 insertions(+), 3 deletions(-)

diff --git a/0Trinitarianism/Quest0.md b/0Trinitarianism/Quest0.md
index 1f11794..6b9ddd9 100644
--- a/0Trinitarianism/Quest0.md
+++ b/0Trinitarianism/Quest0.md
@@ -1,6 +1,7 @@
 # Terms and Types
 
 There are three ways of looking at `A : Type`.
+
   - proof theoretically, '`A` is a proposition'
   - type theoretically, '`A` is a construction'
   - categorically, '`A` is an object in category `Type`'
@@ -8,6 +9,7 @@ There are three ways of looking at `A : Type`.
 A first example of a type construction is the function type.
 Given types `A : Type` and `B : Type`, 
 we have another type `A → B : Type` 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`
@@ -23,12 +25,14 @@ data ⊤ : Type where
 
 It reads '`⊤` is an inductive type with a constructor `tt`',
 with three interpretations
+
   - `⊤` is a proposition and there is a proof of it, called `tt`.
   - `⊤` is a construction with a recipe called `tt`
   - `⊤` is a terminal object: every object has a morphism into `⊤` given by `· ↦ tt`
 
 In general, the expression `a : A` is read '`a` is a term of type `A`',
 and has three interpretations,
+
   - `a` is a proof of the proposition `A`
   - `a` is a recipe for the construction `A`
   - `a` is a generalised element of the object `A` in the category `Type`.
@@ -115,6 +119,7 @@ not _externally_ 'the same'.)
 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 `tt`', or
+
   - the only proof of `⊤` is `tt`
   - the only recipe for `⊤` is `tt`
   - the only one generalized element `tt` in `⊤`
@@ -131,6 +136,7 @@ data ⊥ : Type 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)
@@ -144,8 +150,9 @@ explosion x = { }
 
   - Navigate to the hole and do cases on `x`.
 
-Agda knows that there are no cases so there is nothing to do!
+Agda knows that there are no cases so there is nothing to do (see solutions)!
 This has three interpretations:
+
   - false implies anything (principle of explosion)
   - One can convert recipes of `⊥` to recipes of
     any other construction since
@@ -163,6 +170,7 @@ data ℕ : Type where
 ```
 
 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
@@ -174,6 +182,7 @@ This means it is equipped with morphisms `zero : ⊤ → ℕ`
 and `suc : ℕ → ℕ` such that
 given any `⊤ → A → A` there exist a unique morphism `ℕ → A`
 such that the diagram commutes:
+
 <img src="images/nno.png" 
      alt="nno" 
      width="500"
@@ -208,6 +217,7 @@ These are called _universes_.
 The numberings of universes are called _levels_.
 It will be crucial that types can be treated as terms.
 This will allows us to
+
   - reason about '_structures_' such as 'the structure of a group',
     think 'for all groups'
   - do category theory without stepping out of the theory 
diff --git a/0Trinitarianism/Quest1.md b/0Trinitarianism/Quest1.md
index 024a228..da3bf23 100644
--- a/0Trinitarianism/Quest1.md
+++ b/0Trinitarianism/Quest1.md
@@ -36,9 +36,11 @@ isEven n = ?
   since these are the only constructors given 
   in the definition of `ℕ`."
   This has the following interpretations :
+  
   - propositionally, this is the _principle of mathematical induction_.
   - categorically, this is the universal property of a
     natural numbers object.
+    
 - Navigate to the first hole and check the goal.
   You should see 
   ```
@@ -76,7 +78,8 @@ isEven n = ?
   The reason we have access to the term `isEven n` is again
   because we are in the 'inductive step'.
 - There should now be nothing in the 'agda info' window.
-  This means everything is working.
+  This means everything is working. 
+  (Compare your `isEven` with our [solutions]().)
 
 There are three interpretations of `isEven : ℕ → Type`.
 - Already mentioned, `isEven` is a predicate on `ℕ`.
diff --git a/0Trinitarianism/Quest2.md b/0Trinitarianism/Quest2.md
index 700b3d4..08630c5 100644
--- a/0Trinitarianism/Quest2.md
+++ b/0Trinitarianism/Quest2.md
@@ -1,7 +1,9 @@
 # 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`.
@@ -27,7 +29,7 @@ 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`, 
+- For all even naturals `x` and `y`, 
   their sum is even.
 - `isEven (x .fst + y .fst)` is a construction depending on two recipes
   `x` and `y`.