{-

This file proves a variety of basic results about paths:

- refl, sym, cong and composition of paths. This is used to set up
equational reasoning.

- Transport, subst and functional extensionality

- J and its computation rule (up to a path)

- Σ-types and contractibility of singletons

- Converting PathP to and from a homogeneous path with transp

- Direct definitions of lower h-levels

- Export natural numbers

- Export universe lifting

-}
{-# OPTIONS --safe #-}
module Cubical.Foundations.Prelude where

open import Cubical.Core.Primitives public

infixr 30 _∙_
infix  3 _∎
infixr 2 _≡⟨_⟩_
infixr 2.5 _≡⟨_⟩≡⟨_⟩_

-- Basic theory about paths. These proofs should typically be
-- inlined. This module also makes equational reasoning work with
-- (non-dependent) paths.

private
variable
ℓ' : Level
A : Type
B : A  Type
x y z w : A

refl : x  x
refl {x = x} = λ _  x
{-# INLINE refl #-}

sym : x  y  y  x
sym p i = p (~ i)
{-# INLINE sym #-}

symP : {A : I  Type }  {x : A i0}  {y : A i1}
(p : PathP A x y)  PathP  i  A (~ i)) y x
symP p j = p (~ j)

cong :  (f : (a : A)  B a) (p : x  y)
PathP  i  B (p i)) (f x) (f y)
cong f p i = f (p i)
{-# INLINE cong #-}

cong₂ :  {C : (a : A)  (b : B a)  Type }
(f : (a : A)  (b : B a)  C a b)
(p : x  y)
{u : B x} {v : B y} (q : PathP  i  B (p i)) u v)
PathP  i  C (p i) (q i)) (f x u) (f y v)
cong₂ f p q i = f (p i) (q i)
{-# INLINE cong₂ #-}

{- The most natural notion of homogenous path composition
in a cubical setting is double composition:

x ∙ ∙ ∙ > w
^         ^
p⁻¹ |         | r        ^
|         |        j |
y — — — > z          ∙ — >
q                 i

`p ∙∙ q ∙∙ r` gives the line at the top,
`doubleCompPath-filler p q r` gives the whole square
-}

doubleComp-faces : {x y z w : A } (p : x  y) (r : z  w)
(i : I) (j : I)  Partial (i  ~ i) A
doubleComp-faces p r i j (i = i0) = p (~ j)
doubleComp-faces p r i j (i = i1) = r j

_∙∙_∙∙_ : w  x  x  y  y  z  w  z
(p ∙∙ q ∙∙ r) i =
hcomp (doubleComp-faces p r i) (q i)

doubleCompPath-filler : (p : x  y) (q : y  z) (r : z  w)
PathP  j  p (~ j)  r j) q (p ∙∙ q ∙∙ r)
doubleCompPath-filler p q r j i =
hfill (doubleComp-faces p r i) (inS (q i)) j

-- any two definitions of double composition are equal
compPath-unique :  (p : x  y) (q : y  z) (r : z  w)
(α β : Σ[ s  x  w ] PathP  j  p (~ j)  r j) q s)
α  β
compPath-unique p q r (α , α-filler) (β , β-filler) t
=  i  cb i1 i) ,  j i  cb j i)
where cb : I  I  _
cb j i = hfill  j  λ { (t = i0)  α-filler j i
; (t = i1)  β-filler j i
; (i = i0)  p (~ j)
; (i = i1)  r j })
(inS (q i)) j

{- For single homogenous path composition, we take `p = refl`:

x ∙ ∙ ∙ > z
‖         ^
‖         | r        ^
‖         |        j |
x — — — > y          ∙ — >
q                 i

`q ∙ r` gives the line at the top,
`compPath-filler q r` gives the whole square
-}

_∙_ : x  y  y  z  x  z
p  q = refl ∙∙ p ∙∙ q

compPath-filler : (p : x  y) (q : y  z)  PathP  j  x  q j) p (p  q)
compPath-filler p q = doubleCompPath-filler refl p q

-- We could have also defined single composition by taking `r = refl`:

_∙'_ : x  y  y  z  x  z
p ∙' q = p ∙∙ q ∙∙ refl

compPath'-filler : (p : x  y) (q : y  z)  PathP  j  p (~ j)  z) q (p ∙' q)
compPath'-filler p q = doubleCompPath-filler p q refl

-- It's easy to show that `p ∙ q` also has such a filler:
compPath-filler' : (p : x  y) (q : y  z)  PathP  j  p (~ j)  z) q (p  q)
compPath-filler' {z = z} p q j i =
hcomp  k  λ { (i = i0)  p (~ j)
; (i = i1)  q k
; (j = i0)  q (i  k) })
(p (i  ~ j))
-- Note: We can omit a (j = i1) case here since when (j = i1), the whole expression is
--  definitionally equal to `p ∙ q`. (Notice that `p ∙ q` is also an hcomp.) Nevertheless,
--  we could have given `compPath-filler p q k i` as the (j = i1) case.

-- From this, we can show that these two notions of composition are the same
compPath≡compPath' : (p : x  y) (q : y  z)  p  q  p ∙' q
compPath≡compPath' p q j =
compPath-unique p q refl (p  q  , compPath-filler' p q)
(p ∙' q , compPath'-filler p q) j .fst

-- Heterogeneous path composition and its filler:

compPathP :  {A : I  Type } {x : A i0} {y : A i1} {B_i1 : Type } {B : (A i1)  B_i1} {z : B i1}
PathP A x y  PathP  i  B i) y z  PathP  j  ((λ i  A i)  B) j) x z
compPathP {A = A} {x = x} {B = B} p q i =
comp  j  compPath-filler  i  A i) B j i)
j  λ { (i = i0)  x ;
(i = i1)  q j })
(p i)

compPathP' :  { ℓ'} {A : Type } {B : A  Type ℓ'} {x y z : A} {x' : B x} {y' : B y} {z' : B z} {p : x  y} {q : y  z}
(P : PathP  i  B (p i)) x' y') (Q : PathP  i  B (q i)) y' z')
PathP  i  B ((p  q) i)) x' z'
compPathP' {B = B} {x' = x'} {p = p} {q = q} P Q i =
comp  j  B (compPath-filler p q j i))
j  λ { (i = i0)  x'  ;
(i = i1)  Q j })
(P i)

compPathP-filler :  {A : I  Type } {x : A i0} {y : A i1} {B_i1 : Type } {B : A i1  B_i1} {z : B i1}
(p : PathP A x y) (q : PathP  i  B i) y z)
PathP  j  PathP  k  (compPath-filler  i  A i) B j k)) x (q j)) p (compPathP p q)
compPathP-filler {A = A} {x = x} {B = B} p q j i =
fill  j  compPath-filler  i  A i) B j i)
j  λ { (i = i0)  x ;
(i = i1)  q j })
(inS (p i)) j

compPathP'-filler :  { ℓ'} {A : Type } {B : A  Type ℓ'} {x y z : A} {x' : B x} {y' : B y} {z' : B z} {p : x  y} {q : y  z}
(P : PathP  i  B (p i)) x' y') (Q : PathP  i  B (q i)) y' z')
PathP  j  PathP  i  B (compPath-filler p q j i)) x' (Q j)) P (compPathP' {x = x} {y = y} {x' = x'} {y' = y'} P Q)
compPathP'-filler {B = B} {x' = x'} {p = p} {q = q} P Q j i =
fill  j  B (compPath-filler p q j i))
j  λ { (i = i0)  x'  ;
(i = i1)  Q j })
(inS (P i))
j

_≡⟨_⟩_ : (x : A)  x  y  y  z  x  z
_ ≡⟨ x≡y  y≡z = x≡y  y≡z

≡⟨⟩-syntax : (x : A)  x  y  y  z  x  z
≡⟨⟩-syntax = _≡⟨_⟩_
infixr 2 ≡⟨⟩-syntax
syntax ≡⟨⟩-syntax x  i  B) y = x ≡[ i ]⟨ B  y

≡⟨⟩⟨⟩-syntax : (x y : A)  x  y  y  z  z  w  x  w
≡⟨⟩⟨⟩-syntax x y p q r = p ∙∙ q ∙∙ r
infixr 3 ≡⟨⟩⟨⟩-syntax
syntax ≡⟨⟩⟨⟩-syntax x y B C = x ≡⟨ B ⟩≡ y ≡⟨ C ⟩≡

_≡⟨_⟩≡⟨_⟩_ : (x : A)  x  y  y  z  z  w  x  w
_ ≡⟨ x≡y ⟩≡⟨ y≡z  z≡w = x≡y ∙∙ y≡z ∙∙ z≡w

_∎ : (x : A)  x  x
_  = refl

-- Transport, subst and functional extensionality

-- transport is a special case of transp
transport : {A B : Type }  A  B  A  B
transport p a = transp  i  p i) i0 a

-- Transporting in a constant family is the identity function (up to a
-- path). If we would have regularity this would be definitional.
transportRefl : (x : A)  transport refl x  x
transportRefl {A = A} x i = transp  _  A) i x

transport-filler :  {} {A B : Type } (p : A  B) (x : A)
PathP  i  p i) x (transport p x)
transport-filler p x i = transp  j  p (i  j)) (~ i) x

-- We want B to be explicit in subst
subst : (B : A  Type ℓ') (p : x  y)  B x  B y
subst B p pa = transport  i  B (p i)) pa

subst2 :  {ℓ' ℓ''} {B : Type ℓ'} {z w : B} (C : A  B  Type ℓ'')
(p : x  y) (q : z  w)  C x z  C y w
subst2 B p q b = transport  i  B (p i) (q i)) b

substRefl : (px : B x)  subst B refl px  px
substRefl px = transportRefl px

subst-filler : (B : A  Type ℓ') (p : x  y) (b : B x)
PathP  i  B (p i)) b (subst B p b)
subst-filler B p = transport-filler (cong B p)

funExt : {B : A  I  Type ℓ'}
{f : (x : A)  B x i0} {g : (x : A)  B x i1}
((x : A)  PathP (B x) (f x) (g x))
PathP  i  (x : A)  B x i) f g
funExt p i x = p x i

implicitFunExt : {B : A  I  Type ℓ'}
{f : {x : A}  B x i0} {g : {x : A}  B x i1}
({x : A}  PathP (B x) (f {x}) (g {x}))
PathP  i  {x : A}  B x i) f g
implicitFunExt p i {x} = p {x} i

-- the inverse to funExt (see Functions.FunExtEquiv), converting paths
-- between functions to homotopies; `funExt⁻` is called `happly` and
-- defined by path induction in the HoTT book (see function 2.9.2 in
-- section 2.9)
funExt⁻ : {B : A  I  Type ℓ'}
{f : (x : A)  B x i0} {g : (x : A)  B x i1}
PathP  i  (x : A)  B x i) f g
((x : A)  PathP (B x) (f x) (g x))
funExt⁻ eq x i = eq i x

-- J for paths and its computation rule

module _ (P :  y  x  y  Type ℓ') (d : P x refl) where
J : (p : x  y)  P y p
J p = transport  i  P (p i)  j  p (i  j))) d

JRefl : J refl  d
JRefl = transportRefl d

J-∙ : (p : x  y) (q : y  z)
J (p  q)  transport  i  P (q i)  j  compPath-filler p q i j)) (J p)
J-∙ p q k =
transp
i  P (q (i  ~ k))
j  compPath-filler p q (i  ~ k) j)) (~ k)
(J  j  compPath-filler p q (~ k) j))

-- Converting to and from a PathP

module _ {A : I  Type } {x : A i0} {y : A i1} where
toPathP : transp (\ i  A i) i0 x  y  PathP A x y
toPathP p i = hcomp  j  λ { (i = i0)  x
; (i = i1)  p j })
(transp  j  A (i  j)) (~ i) x)

fromPathP : PathP A x y  transp (\ i  A i) i0 x  y
fromPathP p i = transp  j  A (i  j)) i (p i)

-- Whiskering a dependent path by a path

_◁_ :  {} {A : I  Type } {a₀ a₀' : A i0} {a₁ : A i1}
a₀  a₀'  PathP A a₀' a₁  PathP A a₀ a₁
(p  q) i =
hcomp  j  λ {(i = i0)  p (~ j); (i = i1)  q i1}) (q i)

_▷_ :  {} {A : I  Type } {a₀ : A i0} {a₁ a₁' : A i1}
PathP A a₀ a₁  a₁  a₁'  PathP A a₀ a₁'
(p  q) i =
hcomp  j  λ {(i = i0)  p i0; (i = i1)  q j}) (p i)

-- Direct definitions of lower h-levels

isContr : Type   Type
isContr A = Σ[ x  A ] (∀ y  x  y)

isProp : Type   Type
isProp A = (x y : A)  x  y

isSet : Type   Type
isSet A = (x y : A)  isProp (x  y)

isGroupoid : Type   Type
isGroupoid A =  a b  isSet (Path A a b)

is2Groupoid : Type   Type
is2Groupoid A =  a b  isGroupoid (Path A a b)

-- Contractibility of singletons

singlP : (A : I  Type ) (a : A i0)  Type _
singlP A a = Σ[ x  A i1 ] PathP A a x

singl : (a : A)  Type _
singl {A = A} a = singlP  _  A) a

isContrSingl : (a : A)  isContr (singl a)
isContrSingl a = (a , refl) , λ p i  p .snd i , λ j  p .snd (i  j)

isContrSinglP : (A : I  Type ) (a : A i0)  isContr (singlP A a)
isContrSinglP A a .fst = _ , transport-filler  i  A i) a
isContrSinglP A a .snd (x , p) i =
_ , λ j  fill (\ i  A i)  j  λ {(i = i0)  transport-filler  i  A i) a j; (i = i1)  p j}) (inS a) j

SquareP :
(A : I  I  Type )
{a₀₀ : A i0 i0} {a₀₁ : A i0 i1} (a₀₋ : PathP  j  A i0 j) a₀₀ a₀₁)
{a₁₀ : A i1 i0} {a₁₁ : A i1 i1} (a₁₋ : PathP  j  A i1 j) a₁₀ a₁₁)
(a₋₀ : PathP  i  A i i0) a₀₀ a₁₀) (a₋₁ : PathP  i  A i i1) a₀₁ a₁₁)
Type
SquareP A a₀₋ a₁₋ a₋₀ a₋₁ = PathP  i  PathP  j  A i j) (a₋₀ i) (a₋₁ i)) a₀₋ a₁₋

Square :
{a₀₀ a₀₁ : A} (a₀₋ : a₀₀  a₀₁)
{a₁₀ a₁₁ : A} (a₁₋ : a₁₀  a₁₁)
(a₋₀ : a₀₀  a₁₀) (a₋₁ : a₀₁  a₁₁)
Type _
Square a₀₋ a₁₋ a₋₀ a₋₁ = PathP  i  a₋₀ i  a₋₁ i) a₀₋ a₁₋

isSet' : Type   Type
isSet' A =
{a₀₀ a₀₁ : A} (a₀₋ : a₀₀  a₀₁)
{a₁₀ a₁₁ : A} (a₁₋ : a₁₀  a₁₁)
(a₋₀ : a₀₀  a₁₀) (a₋₁ : a₀₁  a₁₁)
Square a₀₋ a₁₋ a₋₀ a₋₁

Cube :
{a₀₀₀ a₀₀₁ : A} {a₀₀₋ : a₀₀₀  a₀₀₁}
{a₀₁₀ a₀₁₁ : A} {a₀₁₋ : a₀₁₀  a₀₁₁}
{a₀₋₀ : a₀₀₀  a₀₁₀} {a₀₋₁ : a₀₀₁  a₀₁₁}
(a₀₋₋ : Square a₀₀₋ a₀₁₋ a₀₋₀ a₀₋₁)
{a₁₀₀ a₁₀₁ : A} {a₁₀₋ : a₁₀₀  a₁₀₁}
{a₁₁₀ a₁₁₁ : A} {a₁₁₋ : a₁₁₀  a₁₁₁}
{a₁₋₀ : a₁₀₀  a₁₁₀} {a₁₋₁ : a₁₀₁  a₁₁₁}
(a₁₋₋ : Square a₁₀₋ a₁₁₋ a₁₋₀ a₁₋₁)
{a₋₀₀ : a₀₀₀  a₁₀₀} {a₋₀₁ : a₀₀₁  a₁₀₁}
(a₋₀₋ : Square a₀₀₋ a₁₀₋ a₋₀₀ a₋₀₁)
{a₋₁₀ : a₀₁₀  a₁₁₀} {a₋₁₁ : a₀₁₁  a₁₁₁}
(a₋₁₋ : Square a₀₁₋ a₁₁₋ a₋₁₀ a₋₁₁)
(a₋₋₀ : Square a₀₋₀ a₁₋₀ a₋₀₀ a₋₁₀)
(a₋₋₁ : Square a₀₋₁ a₁₋₁ a₋₀₁ a₋₁₁)
Type _
Cube a₀₋₋ a₁₋₋ a₋₀₋ a₋₁₋ a₋₋₀ a₋₋₁ =
PathP  i  Square (a₋₀₋ i) (a₋₁₋ i) (a₋₋₀ i) (a₋₋₁ i)) a₀₋₋ a₁₋₋

isGroupoid' : Type   Type
isGroupoid' A =
{a₀₀₀ a₀₀₁ : A} {a₀₀₋ : a₀₀₀  a₀₀₁}
{a₀₁₀ a₀₁₁ : A} {a₀₁₋ : a₀₁₀  a₀₁₁}
{a₀₋₀ : a₀₀₀  a₀₁₀} {a₀₋₁ : a₀₀₁  a₀₁₁}
(a₀₋₋ : Square a₀₀₋ a₀₁₋ a₀₋₀ a₀₋₁)
{a₁₀₀ a₁₀₁ : A} {a₁₀₋ : a₁₀₀  a₁₀₁}
{a₁₁₀ a₁₁₁ : A} {a₁₁₋ : a₁₁₀  a₁₁₁}
{a₁₋₀ : a₁₀₀  a₁₁₀} {a₁₋₁ : a₁₀₁  a₁₁₁}
(a₁₋₋ : Square a₁₀₋ a₁₁₋ a₁₋₀ a₁₋₁)
{a₋₀₀ : a₀₀₀  a₁₀₀} {a₋₀₁ : a₀₀₁  a₁₀₁}
(a₋₀₋ : Square a₀₀₋ a₁₀₋ a₋₀₀ a₋₀₁)
{a₋₁₀ : a₀₁₀  a₁₁₀} {a₋₁₁ : a₀₁₁  a₁₁₁}
(a₋₁₋ : Square a₀₁₋ a₁₁₋ a₋₁₀ a₋₁₁)
(a₋₋₀ : Square a₀₋₀ a₁₋₀ a₋₀₀ a₋₁₀)
(a₋₋₁ : Square a₀₋₁ a₁₋₁ a₋₀₁ a₋₁₁)
Cube a₀₋₋ a₁₋₋ a₋₀₋ a₋₁₋ a₋₋₀ a₋₋₁

-- Essential consequences of isProp and isContr
isProp→PathP :  {B : I  Type }  ((i : I)  isProp (B i))
(b0 : B i0) (b1 : B i1)
PathP  i  B i) b0 b1
isProp→PathP hB b0 b1 = toPathP (hB _ _ _)

isPropIsContr : isProp (isContr A)
isPropIsContr (c0 , h0) (c1 , h1) j =
h0 c1 j , λ y i  hcomp  k  λ { (i = i0)  h0 (h0 c1 j) k;
(i = i1)  h0 y k;
(j = i0)  h0 (h0 y i) k;
(j = i1)  h0 (h1 y i) k}) c0

isContr→isProp : isContr A  isProp A
isContr→isProp (x , p) a b i =
hcomp  j  λ { (i = i0)  p a j
; (i = i1)  p b j }) x

isProp→isSet : isProp A  isSet A
isProp→isSet h a b p q j i =
hcomp  k  λ { (i = i0)  h a a k
; (i = i1)  h a b k
; (j = i0)  h a (p i) k
; (j = i1)  h a (q i) k }) a

isProp→isSet' : isProp A  isSet' A
isProp→isSet' h {a} p q r s i j =
hcomp  k  λ { (i = i0)  h a (p j) k
; (i = i1)  h a (q j) k
; (j = i0)  h a (r i) k
; (j = i1)  h a (s i) k}) a

isPropIsProp : isProp (isProp A)
isPropIsProp f g i a b = isProp→isSet f a b (f a b) (g a b) i

isPropSingl : {a : A}  isProp (singl a)
isPropSingl = isContr→isProp (isContrSingl _)

isPropSinglP : {A : I  Type } {a : A i0}  isProp (singlP A a)
isPropSinglP = isContr→isProp (isContrSinglP _ _)

-- Universe lifting

record Lift {i j} (A : Type i) : Type (ℓ-max i j) where
constructor lift
field
lower : A

open Lift public

liftExt :  {A : Type } {a b : Lift {} {ℓ'} A}  (lower a  lower b)  a  b
liftExt x i = lift (x i)

doubleCompPath-filler∙ : {a b c d : A} (p : a  b) (q : b  c) (r : c  d)
PathP  i  p i  r (~ i)) (p  q  r) q
doubleCompPath-filler∙ {A = A} {b = b} p q r j i =
hcomp  k  λ { (i = i0)  p j
; (i = i1)  side j k
; (j = i1)  q (i  k)})
(p (j  i))
where
side : I  I  A
side i j =
hcomp  k  λ { (i = i1)  q j
; (j = i0)  b
; (j = i1)  r (~ i  k)})
(q j)

PathP→compPathL : {a b c d : A} {p : a  c} {q : b  d} {r : a  b} {s : c  d}
PathP  i  p i  q i) r s
sym p  r  q  s
PathP→compPathL {p = p} {q = q} {r = r} {s = s} P j i =
hcomp  k  λ { (i = i0)  p (j  k)
; (i = i1)  q (j  k)
; (j = i0)  doubleCompPath-filler∙ (sym p) r q (~ k) i
; (j = i1)  s i })
(P j i)

PathP→compPathR : {a b c d : A} {p : a  c} {q : b  d} {r : a  b} {s : c  d}
PathP  i  p i  q i) r s
r  p  s  sym q
PathP→compPathR {p = p} {q = q} {r = r} {s = s} P j i =
hcomp  k  λ { (i = i0)  p (j  (~ k))
; (i = i1)  q (j  (~ k))
; (j = i0)  r i
; (j = i1)  doubleCompPath-filler∙ p s (sym q) (~ k) i})
(P j i)

-- otherdir

compPathL→PathP : {a b c d : A} {p : a  c} {q : b  d} {r : a  b} {s : c  d}
sym p  r  q  s
PathP  i  p i  q i) r s
compPathL→PathP {p = p} {q = q} {r = r} {s = s} P j i =
hcomp  k  λ { (i = i0)  p (~ k  j)
; (i = i1)  q (~ k  j)
; (j = i0)  doubleCompPath-filler∙ (sym p) r q k i
; (j = i1)  s i})
(P j i)

compPathR→PathP : {a b c d : A} {p : a  c} {q : b  d} {r : a  b} {s : c  d}
r  p  s  sym q
PathP  i  p i  q i) r s
compPathR→PathP {p = p} {q = q} {r = r} {s = s} P j i =
hcomp  k  λ { (i = i0)  p (k  j)
; (i = i1)  q (k  j)
; (j = i0)  r i
; (j = i1)  doubleCompPath-filler∙  p s (sym q) k i})
(P j i)

compPathR→PathP∙∙ : {a b c d : A} {p : a  c} {q : b  d} {r : a  b} {s : c  d}
r  p ∙∙ s ∙∙ sym q
PathP  i  p i  q i) r s
compPathR→PathP∙∙ {p = p} {q = q} {r = r} {s = s} P j i =
hcomp  k  λ { (i = i0)  p (k  j)
; (i = i1)  q (k  j)
; (j = i0)  r i
; (j = i1)  doubleCompPath-filler  p s (sym q) (~ k) i})
(P j i)