TheHoTTGame/Trinitarianism/AsTypes/Quest0.agda

module Trinitarianism.AsTypes.Quest0 where

open import Cubical.Core.Everything hiding (_∨_)
-- ------------------------------

{-
  In this branch,
  we develop the point of view of types as constructions / programs.

  Here is our first construction.
-}
data Unit : Type where
  trivial : Unit
{-
  This reads 'Unit is a type of construction and
  there is a recipe for it, called "trivial"'.

  Here is another construction.
-}

data Empty : Type where

{-
  This says that Empty is a construction and
  there are no recipes for it.
-}

{-
  Given two constructions A and B,
  'converting recipes of A into recipes of B' is itself a type of construction,
  written A → B.
  To give a recipe of A → B,
  we assume a recipe x of A and give a recipe y of B.

  Here is an example demonstrating → in action
-}

UnitToUnit : Unit → Unit
UnitToUnit = {!!}

{-
  * 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

  Goal: Unit
  ————————————————————————————————————————————————————————————
  x : Unit

  * this means you have a proof of Unit 'x : Unit' and you need to give a proof of Unit
  * you can now give it a proof of Unit and press C-c C-SPC to fill the hole

  There is more than one proof (see solutions) - are they the same?
-}

{-
  We can also encode "natural numbers" as a type of construction.
-}
data ℕ : Type where
  zero : ℕ
  suc : ℕ → ℕ
{-
  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.
  '
-}
{-
  Let's write a program that
  "given a recipe n of ℕ, tells us whether it is zero".

  TODO finish this.
-}
isZero : ℕ → Type
isZero zero = {!!}
isZero (suc n) = {!!}

{-
Here's how:
  * when x is zero, we give the proposition Unit
  (try typing it in by writing \top then pressing C-c C-SPC)
  * when x is suc n (i.e. 'n + 1', suc for successor) we give Empty (\bot)
This is technically using induction - see AsTypes.

In general a 'predicate on ℕ' is just a 'function' P : ℕ → Type
-}

{-
You can check if zero is indeed zero by clicking C-c C-n,
which brings up a thing on the bottom saying 'Expression',
and you can type the following
isZero zero
isZero (suc zero)
isZero (suc (suc zero))
...
-}

{-
We can prove that 'there exists a natural number that isZero'
in set theory we might write
  ∃ x ∈ ℕ, x = 0
which in agda noation is
  Σ ℕ isZero

In general if we have predicate P : ℕ → Type we would write
  Σ ℕ P
for
  ∃ x ∈ ℕ, P x

To formulate the result Σ ℕ isZero we need to define
a proof of it
-}
ExistsZero : Σ ℕ isZero
ExistsZero = {!!}

{-
To fill the hole, we need to give a natural and a proof that it is zero.
Agda will give the syntax you need:
  * navigate to the correct hole then refine using C-c C-r
  * there are now two holes - but which is which?
  * navigate to the first holes and type C-c C-,
    - for the first hole it will ask you to give it a natural 'Goal: ℕ'
    - for the second hole it will ask you for a proof that
    whatever you put in the first hole is zero 'Goal: isZero ?0' for example
  * try to fill both holes, using C-c C-SPC as before
  * for the second hole you can try also C-c C-r,
    Agda knows there is an obvious proof!
-}

{-
Let's show 'if all natural numbers are zero then we have a contradiction',
where 'a contradiction' is a proof of Empty.
In maths we would write
  (∀ x ∈ ℕ, x = 0) → Empty
and the agda notation for this is
  ((x : ℕ) → isZero x) → Empty

In general if we have a predicate P : ℕ → Type then we write
  (x : ℕ) → P x
to mean
  ∀ x ∈ ℕ, P x
-}

AllZero→Empty : ((x : ℕ) → isZero x) → Empty
AllZero→Empty = {!!}

{-
Here is how we prove it in maths
  * assume hypothesis h, a proof of (x : ℕ) → isZero x
  * apply the hypothesis h to 1, deducing isZero 1, i.e. we get a proof of isZero 1
  * notice isZero 1 IS Empty

Here is how you can prove it here
  * navigate to the hole and check the goal
  * to assume the hypothesis (x : ℕ) → isZero x,
    type an h in front like so
    AllZero→Empty h = { }
  * now do
    * C-c C-l to load the file
    * navigate to the new hole and check the new goal
  * type h in the hole, type C-c C-r
  * this should give h { }
  * navigate to the new hole and check the Goal
  * Explanation
    * (h x) is a proof of isZero x for each x
    * it's now asking for a natural x such that isZero x is Empty
  * Try filling the hole with 0 and 1 and see what Agda says
-}

{-
Let's try to show the mathematical statement
'any natural n is 0 or not'
but we need a definition of 'or'
-}
data OR (P Q : Type) : Type where
  left : P → OR P Q
  right : Q → OR P Q
{-
This reads
  * Given propositions P and Q we have another proposition P or Q
  * There are two ways of proving P or Q
    * given a proof of P, left sends this to a proof of P or Q
    * given a proof of Q, right sends this to a proof of P or Q

Agda supports nice notation using underscores.
-}

data _∨_ (P Q : Type) : Type where
  left : P → P ∨ Q
  right : Q → P ∨ Q

{-
[Important note]
Agda is sensitive to spaces so these are bad

data _ ∨ _ (P Q : Type) : Type where
  left : P → P ∨ Q
  right : Q → P ∨ Q

data _∨_ (P Q : Type) : Type where
  left : P → P∨Q
  right : Q → P∨Q

it is also sensitive to indentation so these are also bad

data _∨_ (P Q : Type) : Type where
left : P → P ∨ Q
right : Q → P ∨ Q

-}

{-
Now we can prove it!
This technically uses induction - see AsTypes.
Fill the missing part of the theorem statement.
You need to first uncomment this by getting rid of the -- in front (C-x C-;)
-}
-- DecidableIsZero : (n : ℕ) → {!!}
-- DecidableIsZero zero = {!!}
-- DecidableIsZero (suc n) = {!!}