amelia
/
cubical


								-- We begin by adding some primitive bindings using the PRIMITIVE pragma.

								--

								-- It goes like this: PRIMITIVE primName varName.

								--

								-- If the varName is dropped, then it's taken to be the same as primName.

								--

								-- If there is a previous declaration for the varName, then the type

								-- is checked against the internally-known "proper" type for the primitive.


								-- Universe of fibrant types

								{-# PRIMITIVE Type #-}


								-- Universe of non-fibrant types

								{-# PRIMITIVE Pretype #-}


								-- Fibrant is a fancy word for "has a composition structure". Most types

								-- we inherit from MLTT are fibrant:

								--

								-- Stuff like products Π, sums Σ, naturals, booleans, lists, etc., all

								-- have composition structures.

								--

								-- The non-fibrant types are part of the structure of cubical

								-- categories: The interval, partial elements, cubical subtypes, ...


								-- The interval

								---------------


								-- The interval has two endpoints i0 and i1.

								-- These form a de Morgan algebra.

								I : Pretype

								{-# PRIMITIVE Interval I #-}


								i0, i1 : I

								{-# PRIMITIVE i0 #-}

								{-# PRIMITIVE i1 #-}


								-- "minimum" on the interval

								iand : I -> I -> I

								{-# PRIMITIVE iand #-}


								-- "maximum" on the interval.

								ior : I -> I -> I

								{-# PRIMITIVE ior #-}


								-- The interpretation of iand as min and ior as max justifies the fact that

								-- ior i (inot i) != i1, since that equality only holds for the endpoints.


								-- inot i = 1 - i is a de Morgan involution.

								inot : I -> I

								{-# PRIMITIVE inot #-}


								-- Paths

								--------


								-- Since every function in type theory is internally continuous,

								-- and the two endpoints i0 and i1 are equal, we can take the type of

								-- equalities to be continuous functions out of the interval.

								-- That is, x ≡ y iff. ∃ f : I -> A, f i0 = x, f i1 = y.


								-- The type PathP generalises this to dependent products (i : I) -> A i.


								PathP : (A : I -> Type) -> A i0 -> A i1 -> Type

								{-# PRIMITIVE PathP #-}


								-- By taking the first argument to be constant we get the equality type

								-- Path.


								Path : {A : Type} -> A -> A -> Type

								Path {A} = PathP (\i -> A)


								-- reflexivity is given by constant paths


								refl : {A : Type} {x : A} -> Path x x

								refl {A} {x} i = x


								-- Symmetry (for dpeendent paths) is given by inverting the argument to the path, such that

								--   sym p i0 = p (inot i0) = p i1

								--   sym p i1 = p (inot i1) = p i0

								-- This has the correct endpoints.


								sym : {A : I -> Type} {x : A i0} {y : A i1} -> PathP A x y -> PathP (\i -> A (inot i)) y x

								sym p i = p (inot i)


								id : {A : Type} -> A -> A

								id x = x


								the : (A : Type) -> A -> A

								the A x = x


								-- The eliminator for the interval says that if you have x : A i0 and y : A i1,

								-- and x ≡ y, then you can get a proof A i for every element of the interval.

								iElim : {A : I -> Type} {x : A i0} {y : A i1} -> PathP A x y -> (i : I) -> A i

								iElim p i = p i


								-- This corresponds to the elimination principle for the HIT

								--   data I : Pretype where

								--     i0 i1 : I

								--     seg   : i0 ≡ i1


								-- The singleton subtype of A at x is the type of elements of y which

								-- are equal to x.

								Singl : (A : Type) -> A -> Type

								Singl A x = (y : A) * Path x y


								-- Contractible types are those for which there exists an element to which

								-- all others are equal.

								isContr : Type -> Type

								isContr A = (x : A) * ((y : A) -> Path x y)


								-- Using the connection \i j -> y.2 (iand i j), we can prove that

								-- singletons are contracible. Together with transport later on,

								-- we get the J elimination principle of paths.

								singContr : {A : Type} {a : A} -> isContr (Singl A a)

								singContr {A} {a} = ((a, \i -> a), \y i -> (y.2 i, \j -> y.2 (iand i j)))


								-- Some more operations on paths. By rearranging parentheses we get a

								-- proof that the images of equal elements are themselves equal.

								cong : {A : Type} {B : A -> Type} (f : (x : A) -> B x) {x : A} {y : A} (p : Path x y) -> PathP (\i -> B (p i)) (f x) (f y)

								cong f p i = f (p i)


								-- These satisfy definitional equalities, like congComp and congId, which are

								-- propositional in vanilla MLTT.

								congComp : {A : Type} {B : Type} {C : Type}

								           {f : A -> B} {g : B -> C} {x : A} {y : A}

								           (p : Path x y)

								        -> Path (cong g (cong f p)) (cong (\x -> g (f x)) p)

								congComp p = refl


								congId : {A : Type} {x : A} {y : A}

								         (p : Path x y)

								     -> Path (cong (id {A}) p) p

								congId p = refl


								-- Just like rearranging parentheses gives us cong, swapping the value

								-- and interval binders gives us function extensionality.

								funext : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x}

								         (h : (x : A) -> Path (f x) (g x))

								      -> Path f g

								funext h i x = h x i


								-- The proposition associated with an element of the interval

								-------------------------------------------------------------


								-- Associated with every element i : I of the interval, we have the type

								-- IsOne i which is inhabited only when i = i1. In the model, this

								-- corresponds to the map [φ] from the interval cubical set to the

								-- subobject classifier.


								IsOne : I -> Pretype

								{-# PRIMITIVE IsOne #-}


								-- The value itIs1 witnesses the fact that i1 = i1.

								itIs1 : IsOne i1


								{-# PRIMITIVE itIs1 #-}


								-- Partial elements

								-------------------

								--

								-- Since a function I -> A has two endpoints, and a function I -> I -> A

								-- has four endpoints + four functions I -> A as "sides" (obtained by

								-- varying argument while holding the other as a bound variable), we

								-- refer to elements of I^n -> A as "cubes".


								-- This justifies the existence of partial elements, which are, as the

								-- name implies, partial cubes. Namely, a Partial φ A is an element of A

								-- which depends on a proof that IsOne φ.


								Partial : I -> Type -> Pretype

								{-# PRIMITIVE Partial #-}


								-- There is also a dependent version where the type A is itself a

								-- partial element.


								PartialP : (phi : I) -> Partial phi Type -> Pretype

								{-# PRIMITIVE PartialP #-}


								-- Why is Partial φ A not just defined as φ -> A? The difference is that

								-- Partial φ A has an internal representation which definitionally relates

								-- any two partial elements which "agree everywhere", that is, have

								-- equivalent values for every possible assignment of variables which

								-- makes IsOne φ hold.


								-- Cubical Subtypes

								--------------------


								-- Given A : Type, phi : I, and a partial element u : A defined on φ,

								-- we have the type Sub A phi u, notated A[phi -> u] in the output of

								-- the type checker, whose elements are "extensions" of u.


								-- That is, element of A[phi -> u] is an element of A defined everywhere

								-- (a total element), which, when IsOne φ, agrees with u.


								Sub  : (A : Type) (phi : I) -> Partial phi A -> Pretype

								{-# PRIMITIVE Sub #-}


								-- Every total element u : A can be made partial on φ by ignoring the

								-- constraint. Furthermore, this "totally partial" element agrees with

								-- the original total element on φ.

								inS : {A : Type} {phi : I} (u : A) -> Sub A phi (\x -> u)

								{-# PRIMITIVE inS #-}


								-- When IsOne φ, outS {A} {φ} {u} x reduces to u itIs1.

								-- This implements the fact that x agrees with u on φ.

								outS : {A : Type} {phi : I} {u : Partial phi A} -> Sub A phi u -> A

								{-# PRIMITIVE outS #-}


								-- The composition operation

								----------------------------


								-- Now that we have syntax for specifying partial cubes,

								-- and specifying that an element agrees with a partial cube,

								-- we can describe the composition operation.


								comp : (A : I -> Type) {phi : I} (u : (i : I) -> Partial phi (A i)) -> Sub (A i0) phi (u i0) -> A i1

								{-# PRIMITIVE comp #-}


								-- In particular, when φ is a disjunction of the form

								-- (j = 0) || (j = 1), we can draw u as being a pair of lines forming a

								-- "tube", an open square with no floor or roof:

								--

								-- Given u = \j [ (i = i0) -> x, (i = i1) -> q j] on the extent i || ~i,

								-- we draw:

								--

								--          x         q i1

								--          |          |

								-- \j -> x  |          | \j -> q j

								--          |          |

								--          x         q i0

								--

								-- The composition operation says that, as long as we can provide a

								-- "floor" connecting x -- q i0, as a total element of A which, on

								-- phi, extends u i0, then we get the "roof" connecting x and q i1

								-- for free.

								--

								-- If we have a path p : x ≡ y, and q : y ≡ z, then we do get the

								-- "floor", and composition gets us the dotted line:

								--

								--        x..........z

								--        |          |

								--     x  |          | q j

								--        |          |

								--        x----------y

								--             p i


								trans : {A : Type} {x : A} {y : A} {z : A} -> PathP (\i -> A) x y -> PathP (\i -> A) y z -> PathP (\i -> A) x z

								trans {A} {x} p q i =

								  comp (\i -> A)

								    {ior i (inot i)}

								    (\j [ (i = i0) -> x, (i = i1) -> q j ])

								    (inS (p i))


								-- In particular when the formula φ = i0 we get the "opposite face" to a

								-- single point, which corresponds to transport.


								transp : (A : I -> Type) (x : A i0) -> A i1

								transp A x = comp A {i0} (\i [ ]) (inS x)


								-- Since we have the iand operator, we can also derive the *filler* of a cube,

								-- which connects the given face and the output of composition.


								fill : (A : I -> Type) {phi : I} (u : (i : I) -> Partial phi (A i)) -> Sub (A i0) phi (u i0) -> (i : I) -> A i

								fill A {phi} u a0 i =

								    comp (\j -> A (iand i j))

								         {ior phi (inot i)}

								         (\j [ (phi = i1) as p -> u (iand i j) p, (i = i0) -> outS a0 ])

								         (inS (outS a0))


								hfill : {A : Type} {phi : I} (u : (i : I) -> Partial phi A) -> Sub A phi (u i0) -> I -> A

								hfill {A} {phi} u a0 i = fill (\i -> A) {phi} u a0 i


								hcomp : {A : Type} {phi : I} (u : (i : I) -> Partial phi A) -> Sub A phi (u i0) -> A

								hcomp {A} {phi} u a0 = comp (\i -> A) {phi} u a0


								-- For instance, the filler of the previous composition square

								-- tells us that trans p refl = p:


								transRefl : {A : Type} {x : A} {y : A} (p : Path x y) -> Path (trans p refl) p

								transRefl p j i = fill (\i -> A) {ior i (inot i)} (\k [ (i = i0) -> x, (i = i1) -> y ]) (inS (p i)) (inot j)


								-- Reduction of composition

								---------------------------

								--

								-- Composition reduces on the structure of the family A : I -> Type to create

								-- the element a1 : (A i1)[phi -> u i1].

								--

								-- For instance, when filling a cube of functions, the behaviour is to

								-- first transport backwards along the domain, apply the function, then

								-- forwards along the codomain.


								transpFun : {A : Type} {B : Type} {C : Type} {D : Type} (p : Path A B) (q : Path C D)

								         -> (f : A -> C) -> Path (transp (\i -> p i -> q i) f)

								                                 (\x -> transp (\i -> q i) (f (transp (\i -> p (inot i)) x)))

								transpFun p q f = refl


								-- transpDFun : {A : I -> Type} {B : (i : I) -> A i -> Type}

								--           -> (f : (x : A i0) -> B i0 x)

								--           -> Path (transp (\i -> (x : A i) -> B i x) f)

								--                   (\x -> transp (\i -> B i (fill (\j -> A (inot j)) (\k []) (inS x) (inot i))) (f (fill (\j -> A (inot j)) (\k []) (inS x) i1)))

								-- transpDFun f = refl


								-- When considering the more general case of a composition respecing sides,

								-- the outer transport becomes a composition.


								-- Glueing and Univalence

								-------------------------


								-- First, let's get some definitions out of the way.

								--

								-- The *fiber* of a function f : A -> B at a point y : B is the type of

								-- inputs x : A which f takes to y, that is, for which there exists a

								-- path f(x) = y.


								fiber : {A : Type} {B : Type} -> (A -> B) -> B -> Type

								fiber f y = (x : A) * Path (f x) y


								-- An *equivalence* is a function where every fiber is contractible.

								-- That is, for every point in the codomain y : B, there is exactly one

								-- point in the input which f maps to y.


								isEquiv : {A : Type} {B : Type} -> (A -> B) -> Type

								isEquiv {A} {B} f = (y : B) -> isContr (fiber {A} {B} f y)


								-- By extracting this point, which must exist because the fiber is contractible,

								-- we can get an inverse of f:


								inverse : {A : Type} {B : Type} {f : A -> B} -> isEquiv f -> B -> A

								inverse eqv y = (eqv y) .1 .1


								-- We can prove that «inverse eqv» is a section of f:


								section : {A : Type} {B : Type} (f : A -> B) (eqv : isEquiv f) -> Path (\x -> f (inverse eqv x)) id

								section f eqv i y = (eqv y) .1 .2 i


								contr : {A : Type} {phi : I} -> isContr A -> (u : Partial phi A) -> A

								contr {A} {phi} p u = comp (\i -> A) {phi} (\i is1 -> p.2 (u is1) i) (inS (p.1))


								-- Proving that it's also a retraction is left as an exercise to the

								-- reader. We can package together a function and a proof that it's an

								-- equivalence to get a capital-E Equivalence.


								Equiv : (A : Type) (B : Type) -> Type

								Equiv A B = (f : A -> B) * isEquiv {A} {B} f


								-- The identity function is an equivalence between any type A and

								-- itself.

								idEquiv : {A : Type} -> isEquiv (id {A})

								idEquiv y = ((y, \i -> y), \u i -> (u.2 (inot i), \j -> u.2 (ior (inot i) j)))


								-- The glue operation expresses that "extensibility is invariant under

								-- equivalence". Less concisely, the Glue type and its constructor,

								-- glue, let us extend a partial element of a partial type to a total

								-- element of a total type, by "gluing" the partial type T using a

								-- partial equivalence e onto a total type A.


								-- In particular, we have that when φ = i1, Glue A [i1 -> (T, f)] = T.


								primGlue : (A : Type) {phi : I}

								           (T : Partial phi Type)

								           (e : PartialP phi (\o -> Equiv (T o) A))

								        -> Type

								{-# PRIMITIVE Glue primGlue #-}


								-- The glue constructor extends the partial element t : T to a total

								-- element of Glue A [φ -> (T, e)] as long as we have a total im : A

								-- which is the image of f(t).

								--

								-- Agreeing with the condition that Glue A [i1 -> (T, e)] = T,

								-- we have that glue {A} {i1} t im => t.

								prim'glue : {A : Type} {phi : I} {T : Partial phi Type} {e : PartialP phi (\o -> Equiv (T o) A)}

								         -> (t  : PartialP phi T)

								         -> (im : Sub A phi (\o -> (e o).1 (t o)))

								         -> primGlue A T e


								{-# PRIMITIVE glue prim'glue #-}


								-- The unglue operation undoes a glueing. Since when φ = i1,

								-- Glue A [φ -> (T, f)] = T, the argument to primUnglue {A} {i1} ...

								-- will have type T, and so to get back an A we need to apply the

								-- partial equivalence f (defined everywhere).


								primUnglue : {A : Type} {phi : I} {T : Partial phi Type} {e : PartialP phi (\o -> Equiv (T o) A)}

								          -> primGlue A {phi} T e -> A


								{-# PRIMITIVE unglue primUnglue #-}


								-- Diagramatically, i : I |- Glue A [(i \/ ~i) -> (T, e)] can be drawn

								-- as giving us the dotted line in:

								--

								--       T i0 ......... T i1

								--        |              |

								--        |              |

								--   e i0 |~            ~| e i1

								--        |              |

								--        |              |

								--       A i0 --------- A i1

								--                A

								--

								-- Where the the two "e" sides are equivalences, and the bottom side is

								-- the line i : I |- A.

								--

								-- Thus, by choosing a base type, a set of partial types and partial

								-- equivalences, we can make a line between two types (T i0) and (T i1).


								Glue : (A : Type) {phi : I} -> Partial phi ((X : Type) * Equiv X A) -> Type

								Glue A {phi} u = primGlue A {phi} (\o -> (u o).1) (\o -> (u o).2)


								-- For example, we can glue together the type A and the type B as long

								-- as there exists an Equiv A B.

								--

								--        A ............ B

								--        |              |

								--        |              |

								--  equiv |~  ua equiv  ~| idEquiv {B}

								--        |              |

								--        |              |

								--        B ------------ B

								--             \i → B

								--

								univalence : {A : Type} {B : Type} -> Equiv A B -> Path A B

								univalence {A} {B} equiv i =

								  Glue B (\[ (i = i0) -> (A, equiv),

								             (i = i1) -> (B, the B, idEquiv {B}) ])


								-- The fact that this diagram has 2 filled-in B sides explains the

								-- complication in the proof below.

								--

								-- In particular, the actual behaviour of transp (\i -> univalence f i)

								-- (x : A) is not just to apply f x to get a B (the left side), it also

								-- needs to:

								--

								--   * For the bottom side, compose along (\i -> B) (the bottom side)

								--   * For the right side, apply the inverse of the identity, which

								--     is just identity, to get *some* b : B

								--

								-- But that b : B might not agree with the sides of the composition

								-- operation in a more general case, so it composes along (\i -> B)

								-- *again*!

								--

								-- Thus the proof: a simple cubical argument suffices, since

								-- for any composition, its filler connects either endpoints. So

								-- we need to come up with a filler for the bottom and right faces.


								univalenceBeta : {A : Type} {B : Type} (f : Equiv A B) -> Path (transp (\i -> univalence f i)) f.1

								univalenceBeta {A} {B} f i a =

								  let

								    -- The bottom left corner

								    botLeft : B

								    botLeft = transp (\i -> B) (f.1 a)


								    -- The "bottom face" filler connects the bottom-left corner, f.1 a,

								    -- and the bottom-right corner, which is the transport of f.1 a

								    -- along \i -> B.

								    botFace : Path (f.1 a) botLeft

								    botFace i = fill (\i -> B) (\j []) (inS (f.1 a)) i


								    -- The "right face" filler connects the bottom-right corner and the

								    -- upper-right corner, which is again a simple transport along

								    -- \i -> B.

								    rightFace : Path (transp (\i -> B) botLeft) botLeft

								    rightFace i = fill (\i -> B) (\j []) (inS botLeft) (inot i)


								    -- The goal is a path between the bottom-left and top-right corners,

								    -- which we can get by composing (in the path sense) the bottom and

								    -- right faces.

								    goal : Path (transp (\i -> B) botLeft) (f.1 a)

								    goal = trans rightFace (\i -> botFace (inot i))

								  in goal i


								-- The terms univalence + univalenceBeta suffice to prove the "full"

								-- univalence axiom of Voevodsky, as can be seen in the paper

								--

								--   Ian Orton, & Andrew M. Pitts. (2017). Decomposing the Univalence Axiom.

								--

								-- Available freely here: https://arxiv.org/abs/1712.04890v3


								J : {A : Type} {x : A}

								    (P : (y : A) -> Path x y -> Type)

								    (d : P x (\i -> x))

								    {y : A} (p : Path x y)

								 -> P y p

								J P d p = transp (\i -> P (p i) (\j -> p (iand i j))) d


								-- Isomorphisms

								---------------

								--

								-- Since isomorphisms are a much more convenient notion of equivalence

								-- than contractible fibers, it's natural to ask why the CCHM paper, and

								-- this implementation following that, decided on the latter for our

								-- definition of equivalence.


								isIso : {A : Type} -> {B : Type} -> (A -> B) -> Type

								isIso {A} {B} f = (g : B -> A) * ((y : B) -> Path (f (g y)) y) * ((x : A) -> Path (g (f x)) x)


								-- The reason is that the family of types IsIso is not a proposition!

								-- This means that there can be more than one way for a function to be

								-- an equivalence. This is Lemma 4.1.1 of the HoTT book.


								Iso : Type -> Type -> Type

								Iso A B = (f : A -> B) * isIso f


								-- Nevertheless, we can prove that any function with an isomorphism

								-- structure has contractible fibers, using a cubical argument adapted

								-- from CCHM's implementation of cubical type theory:

								--

								-- https://github.com/mortberg/cubicaltt/blob/master/experiments/isoToEquiv.ctt#L7-L55


								IsoToEquiv : {A : Type} {B : Type} -> Iso A B -> Equiv A B

								IsoToEquiv {A} {B} iso = (f, \y -> (fCenter y, fIsCenter y)) where

								  f = iso.1

								  g = iso.2.1

								  s = iso.2.2.1

								  t = iso.2.2.2


								  lemIso : (y : B) (x0 : A) (x1 : A) (p0 : Path (f x0) y) (p1 : Path (f x1) y)

								        -> PathP (\i -> fiber f y) (x0, p0) (x1, p1)

								  lemIso y x0 x1 p0 p1 =

								    let

								      rem0 : Path x0 (g y)

								      rem0 i = comp (\i -> A) (\k [ (i = i0) -> t x0 k, (i = i1) -> g y ]) (inS (g (p0 i)))


								      rem1 : Path x1 (g y)

								      rem1 i = comp (\i -> A) (\k [ (i = i0) -> t x1 k, (i = i1) -> g y ]) (inS (g (p1 i)))


								      p : Path x0 x1

								      p i = comp (\i -> A) (\k [ (i = i0) -> rem0 (inot k), (i = i1) -> rem1 (inot k) ]) (inS (g y))


								      fill0 : I -> I -> A

								      fill0 i j = comp (\i -> A) (\k [ (i = i0) -> t x0 (iand j k)

								                                      , (i = i1) -> g y

								                                      , (j = i0) -> g (p0 i)

								                                      ])

								                            (inS (g (p0 i)))


								      fill1 : I -> I -> A

								      fill1 i j = comp (\i -> A) (\k [ (i = i0) -> t x1 (iand j k)

								                                      , (i = i1) -> g y

								                                      , (j = i0) -> g (p1 i) ])

								                            (inS (g (p1 i)))


								      fill2 : I -> I -> A

								      fill2 i j = comp (\i -> A) (\k [ (i = i0) -> rem0 (ior j (inot k))

								                                      , (i = i1) -> rem1 (ior j (inot k))

								                                      , (j = i1) -> g y ])

								                            (inS (g y))


								      sq : I -> I -> A

								      sq i j = comp (\i -> A) (\k [ (i = i0) -> fill0 j (inot k)

								                                  , (i = i1) -> fill1 j (inot k)

								                                  , (j = i1) -> g y

								                                  , (j = i0) -> t (p i) (inot k) ])

								                          (inS (fill2 i j))


								      sq1 : I -> I -> B

								      sq1 i j = comp (\i -> B) (\k [ (i = i0) -> s (p0 j) k

								                              , (i = i1) -> s (p1 j) k

								                              , (j = i0) -> s (f (p i)) k

								                              , (j = i1) -> s y k

								                              ])

								                          (inS (f (sq i j)))

								    in \i -> (p i, \j -> sq1 i j)


								  fCenter : (y : B) -> fiber f y

								  fCenter y = (g y, s y)


								  fIsCenter : (y : B) (w : fiber f y) -> Path (fCenter y) w

								  fIsCenter y w = lemIso y (fCenter y).1 w.1 (fCenter y).2 w.2


								-- We can prove that any involutive function is an isomorphism, since

								-- such a function is its own inverse.


								involToIso : {A : Type} (f : A -> A) -> ((x : A) -> Path (f (f x)) x) -> isIso f

								involToIso {A} f inv = (f, inv, inv)


								-- An example of univalence

								---------------------------

								--

								-- The classic example of univalence is the equivalence

								-- not : Bool \simeq Bool.

								--

								-- We define it here.


								data Bool : Type where

								  true : Bool

								  false : Bool


								not : Bool -> Bool

								not = \case

								  true -> false

								  false -> true


								elimBool : (P : Bool -> Type) -> P true -> P false -> (b : Bool) -> P b

								elimBool P x y = \case

								  true -> x

								  false -> y


								if : {A : Type} -> A -> A -> Bool -> A

								if x y = \case

								  true -> x

								  false -> y


								-- By pattern matching it suffices to prove (not (not true)) ≡ true and

								-- not (not false) ≡ false. Since not (not true) computes to true (resp.

								-- false), both proofs go through by refl.

								notInvol : (x : Bool) -> Path (not (not x)) x

								notInvol = elimBool (\b -> Path (not (not b)) b) refl refl


								notp : Path Bool Bool

								notp = univalence (IsoToEquiv (not, involToIso not notInvol))


								-- This path actually serves to prove a simple lemma about the universes

								-- of HoTT, namely, that any univalent universe is not a 0-type. If we

								-- had HITs, we could prove that this fact holds for any n, but for now,

								-- proving it's not an h-set is the furthest we can go.


								-- First we define what it means for something to be false. In type theory,

								-- we take ¬P = P → ⊥, where the bottom type is the only type satisfying

								-- the elimination principle

								--

								--    elimBottom : (P : bottom -> Type) -> (b : bottom) -> P b

								--

								-- This follows from setting bottom := ∀ A, A.


								bottom : Type

								bottom = {A : Type} -> A


								elimBottom : (P : bottom -> Type) -> (b : bottom) -> P b

								elimBottom P x = x


								-- We prove that true != false by transporting along the path

								--

								--                    \i -> if (Bool -> Bool) A (p i)

								--   (Bool -> Bool) ------------------------------------ A

								--

								-- To verify that this has the correct endpoints, check out the endpoints

								-- for p:

								--

								--             true ------------------------------------ false

								--

								-- and evaluate the if at either end.


								trueNotFalse : Path true false -> bottom

								trueNotFalse p {A} = transp (\i -> if (Bool -> Bool) A (p i)) id


								-- To be an h-Set is to have no "higher path information". Alternatively,

								--

								-- isHSet A = (x : A) (y : A) -> isHProp (Path x y)

								--

								isHSet : Type -> Type

								isHSet A = {x : A} {y : A} (p : Path x y) (q : Path x y) -> Path p q


								-- We can prove *a* contradiction (note: this is a direct proof!) by adversarially

								-- choosing two paths p, q that we know are not equal. Since "equal" paths have

								-- equal behaviour when transporting, we can choose two paths p, q and a point x

								-- such that transporting x along p gives a different result from x along q.

								--

								-- Since transp notp = not but transp refl = id, that's what we go with. The choice

								-- of false as the point x is just from the endpoints of trueNotFalse.


								universeNotSet : isHSet Type -> bottom

								universeNotSet itIs = trueNotFalse (\i -> transp (\j -> itIs notp refl i j) false)


								-- Funext is an inverse of happly

								---------------------------------

								--

								-- Above we proved function extensionality, namely, that functions

								-- pointwise equal everywhere are themselves equal.

								-- However, this formulation of the axiom is known as "weak" function

								-- extensionality. The strong version is as follows:


								Hom : {A : Type} {B : A -> Type} (f : (x : A) -> B x) -> (g : (x : A) -> B x) -> Type

								Hom {A} f g = (x : A) -> Path (f x) (g x)


								happly : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x}

								      -> (p : Path f g) -> Hom f g

								happly p x i = p i x


								-- Strong function extensionality: happly is an equivalence.


								happlyIsIso : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x}

								           -> isIso {Path f g} {Hom f g} happly

								happlyIsIso {A} {B} {f} {g} = (funext {A} {B} {f} {g}, \hom -> refl, \path -> refl)


								pathIsHom : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x}

								         -> Path (Path f g) (Hom f g)

								pathIsHom {A} {B} {f} {g} =

								  let

								    theIso : Iso (Path f g) (Hom f g)

								    theIso = (happly {A} {B} {f} {g}, happlyIsIso {A} {B} {f} {g})

								  in univalence (IsoToEquiv theIso)


								-- Inductive types

								-------------------

								--

								-- An inductive type is a type freely generated by a finite set of

								-- constructors. For instance, the type of natural numbers is generated

								-- by the constructors for "zero" and "successor".


								data Nat : Type where

								  zero : Nat

								  succ : Nat -> Nat


								-- Pattern matching allows us to prove that these initial types are

								-- initial algebras for their corresponding functors.


								Nat_elim : (P : Nat -> Type) -> P zero -> ((x : Nat) -> P x -> P (succ x)) -> (x : Nat) -> P x

								Nat_elim P pz ps = \case

								  zero -> pz

								  succ x -> ps x (Nat_elim P pz ps x)


								-- The type of integers can be defined as A + B, where "pos n" means +n

								-- and "neg n" means -(n + 1).


								data Int : Type where

								  pos : Nat -> Int

								  neg : Nat -> Int


								-- On this representation we can define the successor and predecessor

								-- functions by (nested) induction.


								sucZ : Int -> Int

								sucZ = \case

								  pos n -> pos (succ n)

								  neg n ->

								    let suc_neg : Nat -> Int

								        suc_neg = \case

								          zero -> pos zero

								          succ n -> neg n

								     in suc_neg n


								predZ : Int -> Int

								predZ = \case

								  pos n ->

								    let pred_pos : Nat -> Int

								        pred_pos = \case

								          zero -> neg zero

								          succ n -> pos n

								     in pred_pos n

								  neg n -> neg (succ n)


								-- And prove that the successor function is an isomorphism, and thus, an

								-- equivalence.


								sucEquiv : isIso sucZ

								sucEquiv =

								  let

								    sucPredZ : (x : Int) -> Path (sucZ (predZ x)) x

								    sucPredZ = \case

								      pos n ->

								        let k : (n : Nat) -> Path (sucZ (predZ (pos n))) (pos n)

								            k = \case

								              zero -> refl

								              succ n -> refl

								         in k n

								      neg n -> refl

								    predSucZ : (x : Int) -> Path (predZ (sucZ x)) x

								    predSucZ = \case

								      pos n -> refl

								      neg n ->

								        let k : (n : Nat) -> Path (predZ (sucZ (neg n))) (neg n)

								            k = \case

								              zero -> refl

								              succ n -> refl

								         in k n

								   in (predZ, sucPredZ, predSucZ)


								-- Univalence gives us a path between integers such that transp intPath

								-- x = suc x, transp (sym intPath) x = pred x


								intPath : Path Int Int

								intPath = univalence (IsoToEquiv (sucZ, sucEquiv))


								-- Higher inductive types

								-------------------------

								--

								-- While inductive types let us generate discrete spaces like the

								-- naturals or integers, they do not support defining higher-dimensional

								-- structures given by spaces with points and paths.


								-- A very simple higher inductive type is the interval, given by


								data Interval : Type where

								  ii0 : Interval

								  ii1 : Interval

								  seg i : Interval [ (i = i0) -> ii0, (i = i1) -> ii1 ]


								-- This expresses that we have two points ii0 and ii1 and a path (\i ->

								-- seg i) with endpoints ii0 and ii1.


								-- With this type we can reproduce the proof of Lemma 6.3.2 from the

								-- HoTT book:


								iFunext : {A : Type} {B : A -> Type} (f : (x : A) -> B x) (g : (x : A) -> B x) -> ((x : A) -> Path (f x) (g x)) -> Path f g

								iFunext f g p i = h' (seg i) where

								  h : (x : A) -> Interval -> B x

								  h x = \case

								    ii0 -> f x

								    ii1 -> g x

								    seg i -> p x i


								  h' : Interval -> (x : A) -> B x

								  h' i x = h x i


								-- Of course, Cubical Type Theory also has an interval (pre)type, but

								-- that, unlike the Interval here, is not Kan: it has no composition

								-- structure.


								-- Another simple higher-inductive type is the circle, with a point and

								-- a non-trivial loop, (\i -> loop i).


								data S1 : Type where

								  base : S1

								  loop i : S1 [ (i = i1) -> base, (i = i0) -> base ]


								-- By writing a function from the circle to the universe of types Type,

								-- we can calculate winding numbers along the circle.


								helix : S1 -> Type

								helix = \case

								  base -> Int

								  loop i -> intPath i


								winding : Path base base -> Int

								winding p = transp (\i -> helix (p i)) (pos zero)


								-- For instance, going around the loop once has a winding number of +1,


								windingLoop : Path (winding (\i -> loop i)) (pos (succ zero))

								windingLoop = refl


								-- Going backwards has a winding number of -1 (remember the

								-- representation of integers),


								windingSymLoop : Path (winding (\i -> loop (inot i))) (neg zero)

								windingSymLoop = refl


								-- And going around the trivial loop (\i -> base) goes around the the

								-- non-trivial loop (\i -> loop) zero times.


								windingBase : Path (winding (\i -> base)) (pos zero)

								windingBase = refl


								-- One particularly general higher inductive type is the homotopy pushout,

								-- which can be seen as a kind of sum B + C with the extra condition that

								-- whenever x and y are in the image of f (resp. g), inl x ≡ inr y.


								data Pushout {A : Type} {B : Type} {C : Type} (f : A -> B) (g : A -> C) : Type where

								  inl : (x : B) -> Pushout f g

								  inr : (y : C) -> Pushout f g

								  push i : (a : A) -> Pushout f g [ (i = i0) -> inl (f a), (i = i1) -> inr (g a) ]


								-- The name is due to the category-theoretical notion of pushout.

								-- TODO: finish writing this tomorrow lol


								data Susp (A : Type) : Type where

								  north : Susp A

								  south : Susp A

								  merid i : A -> Susp A [ (i = i0) -> north, (i = i1) -> south ]


								data Unit : Type where

								  tt : Unit


								poSusp : Type -> Type

								poSusp A = Pushout {A} {Unit} {Unit} (\x -> tt) (\x -> tt)


								poSusp_to_Susp : {A : Type} -> poSusp A -> Susp A

								poSusp_to_Susp = \case

								  inl x -> north

								  inr x -> south

								  push x i -> merid x i


								Susp_to_poSusp : {A : Type} -> Susp A -> poSusp A

								Susp_to_poSusp = \case

								  north -> inl tt

								  south -> inr tt

								  merid x i -> push x i


								Susp_to_poSusp_to_Susp : {A : Type} -> (x : Susp A) -> Path (poSusp_to_Susp (Susp_to_poSusp x)) x

								Susp_to_poSusp_to_Susp = \case

								  north -> refl

								  south -> refl

								  merid x i -> refl


								unitEta : (x : Unit) -> Path x tt

								unitEta = \case tt -> refl


								poSusp_to_Susp_to_poSusp : {A : Type} -> (x : poSusp A) -> Path (Susp_to_poSusp (poSusp_to_Susp x)) x

								poSusp_to_Susp_to_poSusp {A} = \case

								  inl x -> cong inl (sym (unitEta x))

								  inr x -> cong inr (sym (unitEta x))

								  push x i -> refl


								Susp_is_poSusp : {A : Type} -> Path (Susp A) (poSusp A)

								Susp_is_poSusp {A} = univalence (IsoToEquiv (Susp_to_poSusp {A}, poSusp_to_Susp {A}, poSusp_to_Susp_to_poSusp {A}, Susp_to_poSusp_to_Susp {A}))