-- Pullback from product and equalizer
--
--
--                        Shinji KONO <kono@ie.u-ryukyu.ac.jp>
----

open import Category -- https://github.com/konn/category-agda
open import Level
module pullback { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where

open import HomReasoning
open import cat-utility

-- 
-- Pullback
--         f
--     a -------> c
--     ^          ^                 
--  π1 |          |g
--     |          |
--    ab -------> b
--     ^   π2
--     |
--     | equalizer (f π1) (g π1) = ( π1' × π2' )
--     d   
--

open Equalizer
open Product
open Pullback

pullback-from :  (a b c ab d : Obj A) 
      ( f : Hom A a c )    ( g : Hom A b c )
      ( π1 : Hom A ab a )  ( π2 : Hom A ab b ) ( e : Hom A d ab )
      ( eqa : {a b c : Obj A} → (f g : Hom A a b)  → {e : Hom A c a }  → Equalizer A e f g ) 
      ( prod : Product A a b ab π1 π2 ) → Pullback A a b c d f g 
          ( A [  π1 o equalizer ( eqa ( A [ f  o π1 ] ) ( A [ g  o π2 ] ){e} )  ] )
          ( A [  π2 o equalizer ( eqa ( A [ f  o π1 ] ) ( A [ g  o π2 ] ){e} )  ] ) 
pullback-from  a b c ab d f g π1 π2 e eqa prod =  record {
              commute = commute1 ;
              p = p1 ; 
              π1p=π1 = λ {d} {π1'} {π2'} {eq} → π1p=π11  {d} {π1'} {π2'} {eq} ; 
              π2p=π2 = λ {d} {π1'} {π2'} {eq} → π2p=π21  {d} {π1'} {π2'} {eq} ; 
              uniqueness = uniqueness1
      } where 
      commute1 :  A [ A [ f o A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] ≈ A [ g o A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] ]
      commute1 = let open ≈-Reasoning (A) in
             begin
                    f o ( π1 o equalizer (eqa ( f o π1 ) ( g o π2 )) ) 
             ≈⟨ assoc ⟩
                    ( f o  π1 ) o equalizer (eqa ( f o π1 ) ( g o π2 )) 
             ≈⟨ fe=ge (eqa (A [ f o π1 ]) (A [ g o π2 ])) ⟩
                    ( g o  π2 ) o equalizer (eqa ( f o π1 ) ( g o π2 )) 
             ≈↑⟨ assoc ⟩
                    g o ( π2 o equalizer (eqa ( f o π1 ) ( g o π2 )) ) 
             ∎
      lemma1 :  {d' : Obj A} {π1' : Hom A d' a} {π2' : Hom A d' b} → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → 
                      A [ A [ A [ f o π1 ] o (prod × π1') π2' ] ≈ A [ A [ g o π2 ] o (prod × π1') π2' ] ]
      lemma1  {d'} { π1' } { π2' } eq  = let open ≈-Reasoning (A) in 
             begin
                    ( f o π1 ) o (prod × π1') π2'
             ≈↑⟨ assoc ⟩
                     f o ( π1  o (prod × π1') π2' )
             ≈⟨ cdr (π1fxg=f prod)  ⟩
                     f o  π1'
             ≈⟨ eq ⟩
                     g o  π2'
             ≈↑⟨ cdr (π2fxg=g prod)  ⟩
                     g o ( π2  o (prod × π1') π2'  )
             ≈⟨ assoc ⟩
                    ( g o π2 ) o (prod × π1') π2'
             ∎
      p1 :  {d' : Obj A} {π1' : Hom A d' a} {π2' : Hom A d' b} → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → Hom A d' d
      p1 {d'} { π1' } { π2' } eq  = 
         let open ≈-Reasoning (A) in k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod  π1'  π2' ) ( lemma1 eq )
      π1p=π11 :   {d₁ : Obj A} {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → 
            A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π1' ]
      π1p=π11 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in
             begin
                     ( π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq
             ≈⟨⟩
                     ( π1 o e) o  k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod  π1'  π2' ) (lemma1 eq)
             ≈↑⟨ assoc ⟩
                      π1 o ( e o  k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod  π1'  π2' ) (lemma1 eq) )
             ≈⟨ cdr ( ek=h  ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} )) ⟩
                      π1 o  (_×_ prod  π1'  π2' ) 
             ≈⟨ π1fxg=f prod ⟩
                     π1'
             ∎
-- π1fxg=f  prod 
      π2p=π21 : {d₁ : Obj A} {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π2' ]
      π2p=π21  {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in
             begin
                     ( π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq
             ≈⟨⟩
                     ( π2 o e) o  k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod  π1'  π2' ) (lemma1 eq)
             ≈↑⟨ assoc ⟩
                      π2 o ( e o  k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod  π1'  π2' ) (lemma1 eq) )
             ≈⟨ cdr ( ek=h  ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} )) ⟩
                      π2 o  (_×_ prod  π1'  π2' ) 
             ≈⟨ π2fxg=g prod ⟩
                     π2'
             ∎
      uniqueness1 :  {d₁ : Obj A} (p' : Hom A d₁ d) {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} →
        {eq1 : A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π1' ]} →
        {eq2 : A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π2' ]} →
        A [ p1 eq ≈ p' ]
      uniqueness1 = {!!}