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

open import Level using (Level; levelOfTerm)
open import Categories.Category using (Category)

module Category.Semiadditive {o β„“ e : Level} (π’ž : Category o β„“ e) where

import Categories.Morphism.Reasoning π’ž as β‡’-Reasoning
import Categories.Category.Cartesian.SymmetricMonoidal π’ž as CartesianSymmetricMonoidal

open import Algebra using (IsCommutativeMonoid; CommutativeMonoid)
open import Categories.Category.CMonoidEnriched using (CM-Category)
open import Categories.Category.Cartesian π’ž using (Cartesian)
open import Categories.Category.Cartesian.Monoidal using (module CartesianMonoidal)
open import Categories.Category.Cocartesian π’ž using (Cocartesian)
open import Categories.Category.Monoidal.Bundle using (MonoidalCategory; SymmetricMonoidalCategory)
open import Categories.Object.Zero π’ž using (Zero)
open import Category.BinaryBiproducts π’ž using (BinaryBiproducts)
open import Data.Product using (_,_)

private
  module π’ž = Category π’ž
  variable
    A B C D : π’ž.Obj

-- A semiadditive category has all finite biproducts
record Semiadditive : Set (levelOfTerm π’ž) where

  field
    zero : Zero
    biproducts : BinaryBiproducts

  open Zero zero public
  open BinaryBiproducts biproducts public

  open π’ž

  infix 8 _+_

  field
    𝟎 : A β‡’ B
    𝟎-resp-∘ : {h : A β‡’ B} {k : C β‡’ D} β†’ k ∘ 𝟎 ∘ h β‰ˆ 𝟎
    _+_ : A β‡’ B β†’ A β‡’ B β†’ A β‡’ B
    +-resp-β‰ˆ : {x y u v : A β‡’ B} β†’ x β‰ˆ y β†’ u β‰ˆ v β†’ x + u β‰ˆ y + v
    +-resp-∘ : {f g : B β‡’ C} {h : A β‡’ B} {k : C β‡’ D} β†’ k ∘ (f + g) ∘ h β‰ˆ k ∘ f ∘ h + k ∘ g ∘ h
    +-identityΛ‘ : (f : A β‡’ B) β†’ 𝟎 + f β‰ˆ f
    +-identityΚ³ : (f : A β‡’ B) β†’ f + 𝟎 β‰ˆ f

  open HomReasoning
  open β‡’-Reasoning

  +-cong : {x y u v : A β‡’ B} β†’ x β‰ˆ y β†’ u β‰ˆ v β†’ x + u β‰ˆ y + v
  +-cong = +-resp-β‰ˆ

  +-resp-∘ˑ
      : {A B C : Obj}
        {f g : A β‡’ B}
        {k : B β‡’ C}
      β†’ k ∘ (f + g) β‰ˆ k ∘ f + k ∘ g
  +-resp-∘ˑ {f = f} {g} {h} = begin
      h ∘ (f + g)             β‰ˆβŸ¨ refl⟩∘⟨ identityΚ³ ⟨
      h ∘ (f + g) ∘ id        β‰ˆβŸ¨ +-resp-∘ ⟩
      h ∘ f ∘ id + h ∘ g ∘ id β‰ˆβŸ¨ +-resp-β‰ˆ (refl⟩∘⟨ identityΚ³) (refl⟩∘⟨ identityΚ³) ⟩
      h ∘ f + h ∘ g           ∎

  𝟎-resp-∘ˑ
      : {A B C : Obj}
        {f : B β‡’ C}
      β†’ f ∘ 𝟎 {A} {B} β‰ˆ 𝟎
  𝟎-resp-∘ˑ {f = f} = begin
      f ∘ 𝟎       β‰ˆβŸ¨ refl⟩∘⟨ identityΚ³ ⟨
      f ∘ 𝟎 ∘ id  β‰ˆβŸ¨ 𝟎-resp-∘ ⟩
      𝟎           ∎

  module _ {A B : Obj} where

    Ο€β‚βˆ˜iβ‚‚β‰ˆ0 : π₁ {A} {B} ∘ iβ‚‚ β‰ˆ zeroβ‡’
    Ο€β‚βˆ˜iβ‚‚β‰ˆ0 = begin
        π₁ ∘ iβ‚‚             β‰ˆβŸ¨ π₁iβ‚‚-absorbΛ‘ zeroβ‡’ ⟨
        zeroβ‡’ {A} ∘ π₁ ∘ iβ‚‚ β‰ˆβŸ¨ zero-∘ʳ (π₁ ∘ iβ‚‚) ⟩
        zeroβ‡’               ∎

    Ο€β‚‚βˆ˜iβ‚β‰ˆ0 : Ο€β‚‚ {A} {B} ∘ i₁ β‰ˆ zeroβ‡’
    Ο€β‚‚βˆ˜iβ‚β‰ˆ0 = begin
        Ο€β‚‚ ∘ i₁             β‰ˆβŸ¨ Ο€β‚‚i₁-absorbΛ‘ zeroβ‡’ ⟨
        zeroβ‡’ {A} ∘ Ο€β‚‚ ∘ i₁ β‰ˆβŸ¨ zero-∘ʳ (Ο€β‚‚ ∘ i₁) ⟩
        zeroβ‡’               ∎

    πŸŽβ‰ˆzeroβ‡’ : 𝟎 {A} {B} β‰ˆ zeroβ‡’
    πŸŽβ‰ˆzeroβ‡’ = begin
        𝟎         β‰ˆβŸ¨ 𝟎-resp-∘ˑ ⟨
        zeroβ‡’ ∘ 𝟎 β‰ˆβŸ¨ zero-∘ʳ {A} {B} 𝟎 ⟩
        zeroβ‡’     ∎

  module _ {A B : Obj} where

    +-βˆ‡βˆ˜βŸ¨βŸ© : {f g : A β‡’ B} β†’ f + g β‰ˆ βˆ‡ ∘ ⟨ f , g ⟩
    +-βˆ‡βˆ˜βŸ¨βŸ© {f} {g} = begin
        f + g                   β‰ˆβŸ¨ +-resp-β‰ˆ (insertΛ‘ inject₁) (insertΛ‘ injectβ‚‚) ⟩
        βˆ‡ ∘ i₁ ∘ f + βˆ‡ ∘ iβ‚‚ ∘ g β‰ˆβŸ¨ +-resp-∘ˑ ⟨
        βˆ‡ ∘ (i₁ ∘ f + iβ‚‚ ∘ g)   β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-unique left right ⟨
        βˆ‡ ∘ ⟨ f , g ⟩           ∎
      where
        left : π₁ ∘ (i₁ ∘ f + iβ‚‚ ∘ g) β‰ˆ f
        left = begin
            π₁ ∘ (i₁ ∘ f + iβ‚‚ ∘ g)    β‰ˆβŸ¨ +-resp-∘ˑ ⟩
            π₁ ∘ i₁ ∘ f + π₁ ∘ iβ‚‚ ∘ g β‰ˆβŸ¨ +-resp-β‰ˆ (cancelΛ‘ Ο€β‚βˆ˜iβ‚β‰ˆid) (pullΛ‘ Ο€β‚βˆ˜iβ‚‚β‰ˆ0) ⟩
            f + zeroβ‡’ ∘ g             β‰ˆβŸ¨ +-resp-β‰ˆ Equiv.refl (zero-∘ʳ g) ⟩
            f + zeroβ‡’                 β‰ˆβŸ¨ +-resp-β‰ˆ Equiv.refl πŸŽβ‰ˆzeroβ‡’Β βŸ¨
            f + 𝟎                     β‰ˆβŸ¨ +-identityΚ³ f ⟩
            f                         ∎
        right : Ο€β‚‚ ∘ (i₁ ∘ f + iβ‚‚ ∘ g) β‰ˆ g
        right = begin
            Ο€β‚‚ ∘ (i₁ ∘ f + iβ‚‚ ∘ g)    β‰ˆβŸ¨ +-resp-∘ˑ ⟩
            Ο€β‚‚ ∘ i₁ ∘ f + Ο€β‚‚ ∘ iβ‚‚ ∘ g β‰ˆβŸ¨ +-resp-β‰ˆ (pullΛ‘ Ο€β‚‚βˆ˜iβ‚β‰ˆ0) (cancelΛ‘ Ο€β‚‚βˆ˜iβ‚‚β‰ˆid) ⟩
            zeroβ‡’ ∘ f + g             β‰ˆβŸ¨ +-resp-β‰ˆ (zero-∘ʳ f) Equiv.refl ⟩
            zeroβ‡’ + g                 β‰ˆβŸ¨ +-resp-β‰ˆ πŸŽβ‰ˆzeroβ‡’ Equiv.refl ⟨
            𝟎 + g                     β‰ˆβŸ¨ +-identityΛ‘ g ⟩
            g                         ∎

  Ξ”-βŠ• : {X Y : Obj} β†’ Ξ” {X βŠ• Y} β‰ˆ σ₂₃ ∘ Ξ” ×₁ Ξ”
  Ξ”-βŠ• {X} {Y} = begin
      ⟨ id , id ⟩                                     β‰ˆβŸ¨ ⟨⟩-congβ‚‚ id×₁id id×₁id ⟨
      ⟨ id ×₁ id , id ×₁ id ⟩                         β‰ˆβŸ¨ ⟨⟩-congβ‚‚ (×₁-congβ‚‚ project₁ project₁) (×₁-congβ‚‚ projectβ‚‚ projectβ‚‚) ⟨
      ⟨ (π₁ ∘ Ξ”) ×₁ (π₁ ∘ Ξ”) , (Ο€β‚‚ ∘ Ξ”) ×₁ (Ο€β‚‚ ∘ Ξ”) ⟩ β‰ˆβŸ¨ ⟨⟩-congβ‚‚ Γ—β‚βˆ˜Γ—β‚ Γ—β‚βˆ˜Γ—β‚ ⟨
      ⟨ π₁ ×₁ π₁ ∘ Ξ” ×₁ Ξ” , Ο€β‚‚ ×₁ Ο€β‚‚ ∘ Ξ” ×₁ Ξ” ⟩       β‰ˆβŸ¨ ⟨⟩∘ ⟨
      σ₂₃ ∘ Ξ” ×₁ Ξ”                                    ∎

  βˆ‡-βŠ• : {X Y : Obj} β†’ βˆ‡ {X βŠ• Y} β‰ˆ βˆ‡ ×₁ βˆ‡ ∘ σ₂₃
  βˆ‡-βŠ• {X} {Y} = begin
      [ id , id ]                                     β‰ˆβŸ¨ []-congβ‚‚ id×₁id id×₁id ⟨
      [ id ×₁ id , id ×₁ id ]                         β‰ˆβŸ¨ []-congβ‚‚ (×₁-congβ‚‚ inject₁ inject₁) (×₁-congβ‚‚ injectβ‚‚ injectβ‚‚) ⟨
      [ (βˆ‡ ∘ i₁) ×₁ (βˆ‡ ∘ i₁) , (βˆ‡ ∘ iβ‚‚) ×₁ (βˆ‡ ∘ iβ‚‚) ] β‰ˆβŸ¨ []-congβ‚‚ Γ—β‚βˆ˜Γ—β‚ Γ—β‚βˆ˜Γ—β‚ ⟨
      [ βˆ‡ ×₁ βˆ‡ ∘ i₁ ×₁ i₁ , βˆ‡ ×₁ βˆ‡ ∘ iβ‚‚ ×₁ iβ‚‚ ]       β‰ˆβŸ¨ ∘[] ⟨
      βˆ‡ ×₁ βˆ‡ ∘ [ i₁ ×₁ i₁ , iβ‚‚ ×₁ iβ‚‚ ]                β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-unique (∘[] β—‹ []-congβ‚‚ Ο€β‚βˆ˜Γ—β‚ Ο€β‚βˆ˜Γ—β‚) (∘[] β—‹ []-congβ‚‚ Ο€β‚‚βˆ˜Γ—β‚ Ο€β‚‚βˆ˜Γ—β‚) ⟨
      βˆ‡ ×₁ βˆ‡ ∘ ⟨ π₁ +₁ π₁ , Ο€β‚‚ +₁ Ο€β‚‚ ⟩                β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-congβ‚‚ (×₁-+₁ π₁ π₁) (×₁-+₁ Ο€β‚‚ Ο€β‚‚) ⟨
      βˆ‡ ×₁ βˆ‡ ∘ σ₂₃                                    ∎

  module _ {A B : Obj} where

    +-assoc : (x y z : A β‡’ B) β†’ (x + y) + z β‰ˆ x + (y + z)
    +-assoc x y z = begin
        (x + y) + z                                 β‰ˆβŸ¨ +-βˆ‡βˆ˜βŸ¨βŸ© ⟩
        βˆ‡ ∘ ⟨ x + y , z ⟩                           β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-congΚ³ +-βˆ‡βˆ˜βŸ¨βŸ© ⟩
        βˆ‡ ∘ ⟨ βˆ‡ ∘ ⟨ x , y ⟩ , z ⟩                   β‰ˆβŸ¨ refl⟩∘⟨ first∘⟨⟩ ⟨
        βˆ‡ ∘ βˆ‡ ×₁ id ∘ ⟨ ⟨ x , y ⟩ , z ⟩             β‰ˆβŸ¨ extendΚ³ βˆ‡-assoc-×₁ ⟩
        βˆ‡ ∘ (id ×₁ βˆ‡ ∘ assocΛ‘) ∘ ⟨ ⟨ x , y ⟩ , z ⟩  β‰ˆβŸ¨ refl⟩∘⟨ pullΚ³ assocˑ∘⟨⟩ ⟩
        βˆ‡ ∘ id ×₁ βˆ‡ ∘ ⟨ x , ⟨ y , z ⟩ ⟩             β‰ˆβŸ¨ refl⟩∘⟨ second∘⟨⟩ ⟩
        βˆ‡ ∘ ⟨ x , βˆ‡ ∘ ⟨ y , z ⟩ ⟩                   β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-congΛ‘ +-βˆ‡βˆ˜βŸ¨βŸ© ⟨
        βˆ‡ ∘ ⟨ x , y + z ⟩                           β‰ˆβŸ¨ +-βˆ‡βˆ˜βŸ¨βŸ© ⟨
        x + (y + z)                                 ∎

    βˆ‡βˆ˜+-swap : {A : Obj} β†’ βˆ‡ ∘ +-swap {A} β‰ˆ βˆ‡
    βˆ‡βˆ˜+-swap = begin
        βˆ‡ ∘ [ iβ‚‚ , i₁ ]     β‰ˆβŸ¨ ∘[] ⟩
        [ βˆ‡ ∘ iβ‚‚ , βˆ‡ ∘ i₁ ] β‰ˆβŸ¨ []-congβ‚‚ injectβ‚‚ inject₁ ⟩
        [ id , id ]         ∎

    swapβˆ˜Ξ” : {A : Obj} β†’ swap {A} ∘ Ξ” β‰ˆ Ξ”
    swapβˆ˜Ξ” = begin
        ⟨ Ο€β‚‚ , π₁ ⟩ ∘ Ξ”     β‰ˆβŸ¨ ⟨⟩∘ ⟩
        ⟨ Ο€β‚‚ ∘ Ξ” , π₁ ∘ Ξ” ⟩ β‰ˆβŸ¨ ⟨⟩-congβ‚‚ projectβ‚‚ project₁ ⟩
        ⟨ id , id ⟩         ∎

    +-comm : (x y : A β‡’ B) β†’ x + y β‰ˆ y + x
    +-comm x y = begin
        x + y                   β‰ˆβŸ¨ +-βˆ‡βˆ˜βŸ¨βŸ© ⟩
        βˆ‡ ∘ ⟨ x , y ⟩           β‰ˆβŸ¨ refl⟩∘⟨ swap∘⟨⟩ ⟨
        βˆ‡ ∘ swap ∘ ⟨ y , x ⟩    β‰ˆβŸ¨ refl⟩∘⟨ swapβ‰ˆ+-swap ⟩∘⟨refl ⟩
        βˆ‡ ∘ +-swap ∘ ⟨ y , x ⟩  β‰ˆβŸ¨ pullΛ‘ βˆ‡βˆ˜+-swap ⟩
        βˆ‡ ∘ ⟨ y , x ⟩           β‰ˆβŸ¨ +-βˆ‡βˆ˜βŸ¨βŸ© ⟨
        y + x                   ∎

    isCM : IsCommutativeMonoid (_β‰ˆ_ {A} {B}) _+_ 𝟎
    isCM = record
        { isMonoid = record
            { isSemigroup = record
                { isMagma = record
                    { isEquivalence = equiv
                    ; βˆ™-cong = +-resp-β‰ˆ
                    }
                ; assoc = +-assoc
                }
            ; identity = +-identityΛ‘ , +-identityΚ³
            }
        ; comm = +-comm
        }

  hom : Obj β†’ Obj β†’ CommutativeMonoid β„“ e
  hom A B = record
      { Carrier = A β‡’ B
      ; _β‰ˆ_ = _β‰ˆ_
      ; _βˆ™_ = _+_
      ; Ρ = 𝟎
      ; isCommutativeMonoid = isCM
      }

  cm-category : CM-Category o β„“ e
  cm-category = record
      { π’ž
      ; Hom = hom
      ; +-resp-∘ = +-resp-∘
      ; 0-resp-∘ = 𝟎-resp-∘
      }

  cartesian : Cartesian
  cartesian = record
      { terminal = terminal
      ; products = binaryProducts
      }

  cocartesian : Cocartesian
  cocartesian = record
      { initial = initial
      ; coproducts = binaryCoproducts
      }

record SemiadditiveHelper : Set (levelOfTerm π’ž) where

  field
    zero : Zero
    biproducts : BinaryBiproducts

  open Zero zero
  open BinaryBiproducts biproducts
  open π’ž
  open HomReasoning
  open β‡’-Reasoning

  module _ {A B : Obj} where

    Ο€β‚βˆ˜iβ‚‚β‰ˆ0 : π₁ {A} {B} ∘ iβ‚‚ β‰ˆ zeroβ‡’
    Ο€β‚βˆ˜iβ‚‚β‰ˆ0 = begin
        π₁ ∘ iβ‚‚             β‰ˆβŸ¨ π₁iβ‚‚-absorbΛ‘ zeroβ‡’ ⟨
        zeroβ‡’ {A} ∘ π₁ ∘ iβ‚‚ β‰ˆβŸ¨ zero-∘ʳ (π₁ ∘ iβ‚‚) ⟩
        zeroβ‡’               ∎

    Ο€β‚‚βˆ˜iβ‚β‰ˆ0 : Ο€β‚‚ {A} {B} ∘ i₁ β‰ˆ zeroβ‡’
    Ο€β‚‚βˆ˜iβ‚β‰ˆ0 = begin
        Ο€β‚‚ ∘ i₁             β‰ˆβŸ¨ Ο€β‚‚i₁-absorbΛ‘ zeroβ‡’ ⟨
        zeroβ‡’ {A} ∘ Ο€β‚‚ ∘ i₁ β‰ˆβŸ¨ zero-∘ʳ (Ο€β‚‚ ∘ i₁) ⟩
        zeroβ‡’               ∎

-- Construct hom-set addition from biproduct
semiadditveHelper : SemiadditiveHelper β†’ Semiadditive
semiadditveHelper sh = record
    { zero = zero
    ; biproducts = biproducts
    ; 𝟎 = zeroβ‡’
    ; 𝟎-resp-∘ = 𝟎-resp-∘
    ; _+_ = _+_
    ; +-resp-β‰ˆ = +-cong
    ; +-resp-∘ = +-resp-∘
    ; +-identityΛ‘ = +-identityΛ‘
    ; +-identityΚ³ = +-identityΚ³
    }
  where
    open SemiadditiveHelper sh
    open Zero zero
    open BinaryBiproducts biproducts
    open π’ž
    open HomReasoning
    open β‡’-Reasoning

    𝟎-resp-∘
        : {A B C D : Obj}
          {h : A β‡’ B}
          {k : C β‡’ D}
        β†’ k ∘ zeroβ‡’ ∘ h β‰ˆ zeroβ‡’
    𝟎-resp-∘ {h = h} {k} = begin
        k ∘ zeroβ‡’ ∘ h β‰ˆβŸ¨ pullΛ‘ (zero-∘ˑ k) ⟩
        zeroβ‡’ ∘ h     β‰ˆβŸ¨ zero-∘ʳ h ⟩
        zeroβ‡’         ∎

    infix 8 _+_

    _+_ : A β‡’ B β†’ A β‡’ B β†’ A β‡’ B
    f + g = βˆ‡ ∘ ⟨ f , g ⟩

    +-cong : {x y u v : A β‡’ B} β†’ x β‰ˆ y β†’ u β‰ˆ v β†’ x + u β‰ˆ y + v
    +-cong eq₁ eqβ‚‚ = refl⟩∘⟨ ⟨⟩-congβ‚‚ eq₁ eqβ‚‚

    +-resp-∘
        : {A B C D : Obj}
          {f g : B β‡’ C}
          {h : A β‡’ B}
          {k : C β‡’ D}
        β†’ k ∘ (f + g) ∘ h β‰ˆ k ∘ f ∘ h + k ∘ g ∘ h
    +-resp-∘ {f = f} {g} {h} {k} = begin
        k ∘ (βˆ‡ ∘ ⟨ f , g ⟩) ∘ h         β‰ˆβŸ¨ refl⟩∘⟨ pullΚ³ ⟨⟩∘ ⟩
        k ∘ βˆ‡ ∘ ⟨ f ∘ h , g ∘ h ⟩       β‰ˆβŸ¨ extendΚ³ β‡’βˆ‡-×₁ ⟩
        βˆ‡ ∘ k ×₁ k ∘ ⟨ f ∘ h , g ∘ h ⟩  β‰ˆβŸ¨ refl⟩∘⟨ Γ—β‚βˆ˜βŸ¨βŸ© ⟩
        βˆ‡ ∘ ⟨ k ∘ f ∘ h , k ∘ g ∘ h ⟩   ∎

    +-identityΛ‘ : (x : A β‡’ B) β†’ zeroβ‡’ + x β‰ˆ x
    +-identityΛ‘ x = begin
        βˆ‡ ∘ ⟨ zeroβ‡’ , x ⟩               β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-congβ‚‚ (zero-∘ʳ x) identityΛ‘ ⟨
        βˆ‡ ∘ ⟨ zeroβ‡’ ∘ x , id ∘ x ⟩      β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩∘ ⟨
        βˆ‡ ∘ ⟨ zeroβ‡’ , id ⟩ ∘ x          β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-congβ‚‚ Ο€β‚βˆ˜iβ‚‚β‰ˆ0 Ο€β‚‚βˆ˜iβ‚‚β‰ˆid ⟩∘⟨refl ⟨
        βˆ‡ ∘ ⟨ π₁ ∘ iβ‚‚ , Ο€β‚‚ ∘ iβ‚‚ ⟩ ∘ x   β‰ˆβŸ¨ refl⟩∘⟨ g-Ξ· ⟩∘⟨refl ⟩
        βˆ‡ ∘ iβ‚‚ ∘ x                      β‰ˆβŸ¨ cancelΛ‘ βˆ‡-identityΛ‘ ⟩
        x                               ∎

    +-identityΚ³ : (x : A β‡’ B) β†’ x + zeroβ‡’ β‰ˆ x
    +-identityΚ³ x = begin
        βˆ‡ ∘ ⟨ x , zeroβ‡’ ⟩               β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-congβ‚‚ identityΛ‘ (zero-∘ʳ x) ⟨
        βˆ‡ ∘ ⟨ id ∘ x , zeroβ‡’ ∘ x ⟩      β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩∘ ⟨
        βˆ‡ ∘ ⟨ id , zeroβ‡’ ⟩ ∘ x          β‰ˆβŸ¨ refl⟩∘⟨ ⟨⟩-congβ‚‚ Ο€β‚βˆ˜iβ‚β‰ˆid Ο€β‚‚βˆ˜iβ‚β‰ˆ0 ⟩∘⟨refl ⟨
        βˆ‡ ∘ ⟨ π₁ ∘ i₁ , Ο€β‚‚ ∘ i₁ ⟩ ∘ x   β‰ˆβŸ¨ refl⟩∘⟨ g-Ξ· ⟩∘⟨refl ⟩
        βˆ‡ ∘ i₁ ∘ x                      β‰ˆβŸ¨ cancelΛ‘ βˆ‡-identityΚ³ ⟩
        x                               ∎
generated by cgit v1.2.3 (git 2.39.1) at 2026-10-21 01:08:54 +0000