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 -> Pretype) -> A i0 -> A i1 -> Type

{-# PRIMITIVE PathP #-}


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

-- Path.


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

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


-- reflexivity is given by constant paths


refl : {A : Pretype} {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 -> Pretype} {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 : Pretype) -> 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 -> Pretype} {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


-- Furthermore, if either of i or j are one, then so is (i or j).

isOneL : {i : I} {j : I} -> IsOne i -> IsOne (ior i j)

isOneR : {i : I} {j : I} -> IsOne j -> IsOne (ior i j)


{-# PRIMITIVE itIs1 #-}

{-# PRIMITIVE isOneL #-}

{-# PRIMITIVE isOneR #-}


-- 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 (\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))

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

 (inS (outS 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


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

-- the outer transport becomes a composition.