{-

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 --cubical --no-import-sorts --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 ℓ' : Level
A : Type ℓ
B : A → Type ℓ
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 ℓ} → {x : A i0} → {y : A i1} →
(p : PathP A x y) → PathP (λ i → A (~ i)) y x
symP p j = p (~ j)

cong : ∀ (f : (a : A) → B a) (p : x ≡ y) →
PathP (λ 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 ℓ} →
(f : (a : A) → (b : B a) → C a b) →
(p : x ≡ y) →
{u : B x} {v : B y} (q : PathP (λ i → B (p i)) u v) →
PathP (λ 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 (λ 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 (λ j → p (~ j) ≡ r j) q s)
→ α ≡ β
compPath-unique p q r (α , α-filler) (β , β-filler) t
= (λ i → cb i1 i) , (λ j i → cb j i)
where cb : I → I → _
cb j i = hfill (λ 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 (λ 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 (λ 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 (λ j → p (~ j) ≡ z) q (p ∙ q)
compPath-filler' {z = z} p q j i =
hcomp (λ 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 ℓ} {x : A i0} {y : A i1} {B_i1 : Type ℓ} {B : (A i1) ≡ B_i1} {z : B i1}
→ PathP A x y → PathP (λ i → B i) y z → PathP (λ j → ((λ i → A i) ∙ B) j) x z
compPathP {A = A} {x = x} {B = B} p q i =
comp (λ j → compPath-filler (λ i → A i) B j i)
(λ j → λ { (i = i0) → x ;
(i = i1) → q j })
(p i)

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

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

compPathP'-filler : ∀ {ℓ ℓ'} {A : Type ℓ} {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 (λ i → B (p i)) x' y') (Q : PathP (λ i → B (q i)) y' z')
→ PathP (λ j → PathP (λ 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 (λ j → B (compPath-filler p q j i))
(λ 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 (λ 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 ℓ} → A ≡ B → A → B
transport p a = transp (λ 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 (λ _ → A) i x

transport-filler : ∀ {ℓ} {A B : Type ℓ} (p : A ≡ B) (x : A)
→ PathP (λ i → p i) x (transport p x)
transport-filler p x i = transp (λ 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 (λ 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 (λ i → B (p i) (q i)) b

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

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 (λ 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 (λ 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 (λ 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 (λ i → P (p i) (λ j → p (i ∧ j))) d

JRefl : J refl ≡ d
JRefl = transportRefl d

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

-- Converting to and from a PathP

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

fromPathP : PathP A x y → transp (\ i → A i) i0 x ≡ y
fromPathP p i = transp (λ j → A (i ∨ j)) i (p i)

-- Whiskering a dependent path by a path

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

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

-- Direct definitions of lower h-levels

isContr : Type ℓ → Type ℓ
isContr A = Σ[ x ∈ A ] (∀ y → x ≡ y)

isProp : Type ℓ → Type ℓ
isProp A = (x y : A) → x ≡ y

isSet : Type ℓ → Type ℓ
isSet A = (x y : A) → isProp (x ≡ y)

isGroupoid : Type ℓ → Type ℓ
isGroupoid A = ∀ a b → isSet (Path A a b)

is2Groupoid : Type ℓ → Type ℓ
is2Groupoid A = ∀ a b → isGroupoid (Path A a b)

-- Contractibility of singletons

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

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

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

isSet' : Type ℓ → Type ℓ
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 (λ i → Square (a₋₀₋ i) (a₋₁₋ i) (a₋₋₀ i) (a₋₋₁ i)) a₀₋₋ a₁₋₋

isGroupoid' : Type ℓ → Type ℓ
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 ℓ} → ((i : I) → isProp (B i))
→ (b0 : B i0) (b1 : B i1)
→ PathP (λ i → B i) b0 b1
isProp→PathP hB b0 b1 = toPathP (hB _ _ _)

isPropIsContr : isProp (isContr A)
isPropIsContr z0 z1 j =
( z0 .snd (z1 .fst) j
, λ x i → hcomp (λ k → λ { (i = i0) → z0 .snd (z1 .fst) j
; (i = i1) → z0 .snd x (j ∨ k)
; (j = i0) → z0 .snd x (i ∧ k)
; (j = i1) → z1 .snd x i })
(z0 .snd (z1 .snd x i) j))

isContr→isProp : isContr A → isProp A
isContr→isProp (x , p) a b i =
hcomp (λ 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 (λ 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 (λ 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 ℓ} {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 ℓ} {a b : Lift {ℓ} {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 (λ i → p i ≡ r (~ i)) (p ∙ q ∙ r) q
doubleCompPath-filler∙ {A = A} {b = b} p q r j i =
hcomp (λ 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 (λ 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 (λ 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 (λ 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 (λ 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 (λ 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 (λ i → p i ≡ q i) r s
compPathL→PathP {p = p} {q = q} {r = r} {s = s} P j i =
hcomp (λ 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 (λ i → p i ≡ q i) r s
compPathR→PathP {p = p} {q = q} {r = r} {s = s} P j i =
hcomp (λ 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)