{-# OPTIONS --without-K #-}

open import library.Basics

module library.types.Paths where

module _ {i} {A : Type i} {x y : A} where

  !-equiv : (x == y)  (y == x)
  !-equiv = equiv ! ! !-! !-!

module _ {i j} {A : Type i} {B : Type j} {f : A  B} {b : B} where

  ↓-app=cst-in : {x y : A} {p : x == y} {u : f x == b} {v : f y == b}
     u == (ap f p  v)
     (u == v [  x  f x == b)  p ])
  ↓-app=cst-in {p = idp} q = q

  ↓-app=cst-out : {x y : A} {p : x == y} {u : f x == b} {v : f y == b}
     (u == v [  x  f x == b)  p ])
     u == (ap f p  v)
  ↓-app=cst-out {p = idp} r = r

  ↓-app=cst-eqv : {x y : A} {p : x == y} {u : f x == b} {v : f y == b}
     (u == (ap f p  v))  (u == v [  x  f x == b)  p ])
  ↓-app=cst-eqv {p = idp} = equiv ↓-app=cst-in ↓-app=cst-out 
                             
      _  idp)  _  idp)

  ↓-cst=app-in : {x y : A} {p : x == y} {u : b == f x} {v : b == f y}
     (u ∙' ap f p) == v
     (u == v [  x  b == f x)  p ])
  ↓-cst=app-in {p = idp} idp = idp

  ↓-cst=app-out : {x y : A} {p : x == y} {u : b == f x} {v : b == f y}
     (u == v [  x  b == f x)  p ])
     (u ∙' ap f p) == v
  ↓-cst=app-out {p = idp} idp = idp

module _ {i} {A : Type i} where

  ↓-app=idf-in : {f : A  A} {x y : A} {p : x == y}
    {u : f x == x} {v : f y == y}
     u ∙' p == ap f p  v
     u == v [  z  f z == z)  p ]
  ↓-app=idf-in {p = idp} q = q

  ↓-cst=idf-in : {a : A} {x y : A} {p : x == y} {u : a == x} {v : a == y}
     (u  p) == v
     (u == v [  x  a == x)  p ])
  ↓-cst=idf-in {p = idp} q = ! (∙-unit-r _)  q

  ↓-idf=cst-in : {a : A} {x y : A} {p : x == y} {u : x == a} {v : y == a}
     u == p ∙' v
     (u == v [  x  x == a)  p ])
  ↓-idf=cst-in {p = idp} q = q  ∙'-unit-l _

  ↓-idf=idf-in : {x y : A} {p : x == y} {u : x == x} {v : y == y}
     u  p == p ∙' v
     (u == v [  x  x == x)  p ])
  ↓-idf=idf-in {p = idp} q = ! (∙-unit-r _)  q  ∙'-unit-l _

{- Nondependent identity type -}

↓-='-in :  {i j} {A : Type i} {B : Type j} {f g : A  B}
  {x y : A} {p : x == y} {u : f x == g x} {v : f y == g y}
   (u  ap g p) == (ap f p ∙' v)
   (u == v [  x  f x == g x)  p ])
↓-='-in {p = idp} q = ! (∙-unit-r _)  q  (∙'-unit-l _)

↓-='-out :  {i j} {A : Type i} {B : Type j} {f g : A  B}
  {x y : A} {p : x == y} {u : f x == g x} {v : f y == g y}
   (u == v [  x  f x == g x)  p ])
   (u  ap g p) == (ap f p ∙' v)
↓-='-out {p = idp} q = (∙-unit-r _)  q  ! (∙'-unit-l _)

{- Identity type where the type is dependent -}

↓-=-in :  {i j} {A : Type i} {B : A  Type j} {f g : Π A B}
  {x y : A} {p : x == y} {u : g x == f x} {v : g y == f y}
   (u  apd f p) == (apd g p  v)
   (u == v [  x  g x == f x)  p ])
↓-=-in {B = B} {p = idp} {u} {v} q = ! (◃idp {B = B} u)  q  idp▹ {B = B} v

↓-=-out :  {i j} {A : Type i} {B : A  Type j} {f g : Π A B}
  {x y : A} {p : x == y} {u : g x == f x} {v : g y == f y}
   (u == v [  x  g x == f x)  p ])
   (u  apd f p) == (apd g p  v)
↓-=-out {B = B} {p = idp} {u} {v} q = (◃idp {B = B} u)  q  ! (idp▹ {B = B} v)

-- Dependent path in a type of the form [λ x → g (f x) ≡ x]
module _ {i j} {A : Type i} {B : Type j} (g : B  A) (f : A  B) where
  ↓-∘=idf-in : {x y : A} {p : x == y} {u : g (f x) == x} {v : g (f y) == y}
     ((ap g (ap f p) ∙' v) == (u  p))
     (u == v [  x  g (f x) == x)  p ])
  ↓-∘=idf-in {p = idp} q = ! (∙-unit-r _)  (! q)  (∙'-unit-l _)

-- WIP, derive it from more primitive principles
-- ↓-∘≡id-in f g {p = p} {u} {v} q =
--   ↓-≡-in (u ◃ apd (λ x → g (f x)) p =⟨ apd-∘ f g p |in-ctx (λ t → u ◃ t) ⟩
--         u ◃ ↓-apd-out _ f p (apdd g p (apd f p)) =⟨ apdd-cst (λ _ b → g b) p (ap f p) (! (apd-nd f p)) |in-ctx (λ t → u ◃ ↓-apd-out _ f p t) ⟩
--         u ◃ ↓-apd-out _ f p (apd (λ t → g (π₂ t)) (pair= p (apd f p))) =⟨ apd-∘ π₂ g (pair= p (apd f p)) |in-ctx (λ t → u ◃ ↓-apd-out _ f p t) ⟩
--         u ◃ ↓-apd-out _ f p (↓-apd-out _ π₂ (pair= p (apd f p)) (apdd g (pair= p (apd f p)) (apd π₂ (pair= p (apd f p))))) =⟨ {!!} ⟩
--         apd (λ x → x) p ▹ v ∎)

-- module _ {i j} {A : Type i} {B : Type j} {x y z : A → B} where

--   lhs : 
--     {a a' : A} {p : a == a'} {q : x a == y a} {q' : x a' == y a'}
--     {r : y a == z a} {r' : y a' == z a'}
--     (α : q == q'            [ (λ a → x a == y a) ↓ p ])
--     (β : r ∙ ap z p == ap y p ∙' r')
--     → (q ∙' r) ∙ ap z p == ap x p ∙' q' ∙' r'
--   lhs =
--     (q ∙' r) ∙ ap z p     =⟨ ? ⟩  -- assoc
--     q ∙' (r ∙ ap z p)     =⟨ ? ⟩  -- β
--     q ∙' (ap y p ∙' r')   =⟨ ? ⟩  -- assoc
--     (q ∙' ap y p) ∙' r'   =⟨ ? ⟩  -- ∙ = ∙'
--     (q ∙ ap y p) ∙' r'    =⟨ ? ⟩  -- α
--     (ap x p ∙' q') ∙' r'  =⟨ ? ⟩  -- assoc
--     ap x p ∙' q' ∙' r' ∎
    

--   thing :
--     {a a' : A} {p : a == a'} {q : x a == y a} {q' : x a' == y a'}
--     {r : y a == z a} {r' : y a' == z a'}
--     (α : q == q'            [ (λ a → x a == y a) ↓ p ])
--     (β : r ∙ ap z p == ap y p ∙' r')
--     → (_∙'2ᵈ_ {r = r} {r' = r'} α (↓-='-in β) == ↓-='-in {!!})
--   thing = {!!}