From 0a2cc10ec89a328e932fd3e2736b35a7102740d7 Mon Sep 17 00:00:00 2001
From: Leon Vatthauer <leon.vatthauer@fau.de>
Date: Tue, 5 Sep 2023 18:17:09 +0200
Subject: [PATCH] Worked on defining delay monad correctly

---
 src/Monad/Instance/Delay.lagda.md | 135 ++++++++++++++++++++++++++++--
 1 file changed, 129 insertions(+), 6 deletions(-)

diff --git a/src/Monad/Instance/Delay.lagda.md b/src/Monad/Instance/Delay.lagda.md
index 48a4746..49617b6 100644
--- a/src/Monad/Instance/Delay.lagda.md
+++ b/src/Monad/Instance/Delay.lagda.md
@@ -1,20 +1,24 @@
 <!--
 ```agda
 open import Level
-open import Categories.Category.Core
+open import Categories.Category
+open import Categories.Monad
 open import Categories.Category.Distributive
 open import Categories.Category.Extensive.Bundle
 open import Categories.Category.Extensive
 open import Categories.Category.BinaryProducts
 open import Categories.Category.Cocartesian
 open import Categories.Category.Cartesian
+open import Categories.Category.Cartesian
 open import Categories.Object.Terminal
+open import Categories.Object.Coproduct
 open import Categories.Category.Construction.F-Coalgebras
 open import Categories.Functor.Coalgebra
 open import Categories.Functor
 open import Categories.Functor.Algebra
 open import Categories.Monad.Construction.Kleisli
 open import Categories.Category.Construction.F-Coalgebras
+open import Categories.NaturalTransformation
 import Categories.Morphism as M
 import Categories.Morphism.Reasoning as MR
 ```
@@ -50,6 +54,129 @@ First I postulate the Functor *D*, maybe I should derive it...
 - how to express **Theorem 8** in agda?
 
 ```agda
+
+  -- TODO use this to get i₂ as F-Coalgebra, then use ⊤-id below!!
+  Coalg-cop : ∀ {A B : Obj} → (F : Endofunctor C) → (alg₁ : F-Coalgebra-on F A) → (alg₂ : F-Coalgebra-on F B) → Coproduct (F-Coalgebras F) (record { A = A ; α = alg₁ }) (record { A = B ; α = alg₂ })
+  Coalg-cop {A} {B} F alg₁ alg₂ = record
+    { A+B = record { A = A + B ; α = [ (F₁ i₁ ∘ alg₁) , F₁ i₂ ∘ alg₂ ] }
+    ; i₁ = record { f = i₁ ; commutes = inject₁ }
+    ; i₂ = record { f = i₂ ; commutes = inject₂ }
+    ; [_,_] = λ {CA} h i → record { f = [ F-Coalgebra-Morphism.f h , F-Coalgebra-Morphism.f i ] ; commutes = begin 
+      F-Coalgebra.α CA ∘ [ F-Coalgebra-Morphism.f h , F-Coalgebra-Morphism.f i ] ≈⟨ ∘[] ⟩ 
+      [ F-Coalgebra.α CA ∘ F-Coalgebra-Morphism.f h , F-Coalgebra.α CA ∘ F-Coalgebra-Morphism.f i ] ≈⟨ ⟺ ([]-cong₂ (⟺ (F-Coalgebra-Morphism.commutes h)) ((⟺ (F-Coalgebra-Morphism.commutes i)))) ⟩
+      [ F₁ (F-Coalgebra-Morphism.f h) ∘ alg₁ , F₁ (F-Coalgebra-Morphism.f i) ∘ alg₂ ] ≈⟨ ⟺ ([]-cong₂ ((F-resp-≈ inject₁) ⟩∘⟨refl) ((F-resp-≈ inject₂) ⟩∘⟨refl)) ⟩
+      [ F₁ ([ F-Coalgebra-Morphism.f h , F-Coalgebra-Morphism.f i ] ∘ i₁) ∘ alg₁ , F₁ ([ F-Coalgebra-Morphism.f h , F-Coalgebra-Morphism.f i ] ∘ i₂) ∘ alg₂ ] ≈⟨ ⟺ ([]-cong₂ (pullˡ (⟺ homomorphism)) (pullˡ (⟺ homomorphism))) ⟩
+      [ F₁ ([ F-Coalgebra-Morphism.f h , F-Coalgebra-Morphism.f i ]) ∘ F₁ i₁ ∘ alg₁ , F₁ ([ F-Coalgebra-Morphism.f h , F-Coalgebra-Morphism.f i ]) ∘ F₁ i₂ ∘ alg₂ ] ≈⟨ ⟺ ∘[] ⟩
+      F₁ ([ F-Coalgebra-Morphism.f h , F-Coalgebra-Morphism.f i ]) ∘ [ F₁ i₁ ∘ alg₁ , F₁ i₂ ∘ alg₂ ] ∎ }
+    ; inject₁ = inject₁
+    ; inject₂ = inject₂
+    ; unique = λ eq₁ eq₂ → +-unique eq₁ eq₂
+    }
+    where open Functor F
+
+  delayF : Obj → Endofunctor C
+  delayF Y = record
+    { F₀ = λ X → Y + X
+    ; F₁ = λ f → idC +₁ f
+    ; identity = CC.coproduct.unique id-comm-sym id-comm-sym
+    ; homomorphism = ⟺ (+₁∘+₁ ○ +₁-cong₂ identity² refl) 
+    ; F-resp-≈ = λ eq → +₁-cong₂ refl eq
+    }
+
+  record DelayM : Set (o ⊔ ℓ ⊔ e) where
+    field
+      algebras : ∀ (A : Obj) → Terminal (F-Coalgebras (delayF A))
+    
+    module D A = Functor (delayF A)
+
+    module _ (X : Obj) where
+      open Terminal (algebras X) using (⊤; !)
+      open F-Coalgebra ⊤ using (α) renaming (A to DX)
+
+      -- TODO figure out how to name things...
+      out' : DX ⇒ X + DX
+      out' = α
+
+      D₀ = DX
+
+      out-≅ : DX ≅ X + DX
+      out-≅ = colambek {F = delayF X} (algebras X)
+
+      open _≅_ out-≅ using () renaming (to to out⁻¹; from to out) public
+
+      now : X ⇒ DX
+      now = out⁻¹ ∘ i₁
+
+      later : DX ⇒ DX
+      later = out⁻¹ ∘ i₂
+
+      module _ {Y : Obj} where 
+        coit : Y ⇒ X + Y → Y ⇒ DX
+        coit f = F-Coalgebra-Morphism.f (! {A = record { A = Y ; α = f }})
+
+        coit-commutes : ∀ (f : Y ⇒ X + Y) → out ∘ (coit f) ≈ (idC +₁ coit f) ∘ f
+        coit-commutes f = F-Coalgebra-Morphism.commutes (! {A = record { A = Y ; α = f }})
+    
+    monad : Monad C
+    monad = Kleisli⇒Monad C (record
+      { F₀ = D₀ 
+      ; unit = λ {X} → now X
+      ; extend = extend
+      ; identityʳ = {!   !}
+      ; identityˡ = {!   !}
+      ; assoc = {!   !}
+      ; sym-assoc = {!   !}
+      ; extend-≈ = {!   !}
+      })
+      where
+        module _ {X Y : Obj} (f : X ⇒ D₀ Y) where
+          open Terminal (algebras Y) using (!; ⊤-id)
+          alg : F-Coalgebra (delayF Y)
+          alg = record { A = D₀ X + D₀ Y ; α = (idC +₁ idC +₁ [ idC , idC ]) ∘ (idC +₁ _≅_.to +-assoc ∘ _≅_.to +-comm) ∘ _≅_.to +-assoc ∘ ((out Y ∘ f) +₁ idC) ∘ _≅_.to +-assoc ∘ (out X +₁ idC) }  -- _≅_.to +-assoc
+          extend : D₀ X ⇒ D₀ Y
+          extend = F-Coalgebra-Morphism.f (! {A = alg}) ∘ i₁ {B = D₀ Y}
+          +-assocˡ∘i₁∘i₁ : +-assocˡ ∘ i₁ ∘ i₁ ≈ i₁
+          +-assocˡ∘i₁∘i₁ = begin +-assocˡ ∘ i₁ ∘ i₁ ≈⟨ pullˡ inject₁ ⟩ [ i₁ , i₂ ∘ i₁ ] ∘ i₁ ≈⟨ inject₁ ⟩ i₁ ∎
+          -- i₂-morphism = record { f = i₂ ; commutes = {!   !} }
+          i₂∘! : F-Coalgebra-Morphism.f (! {A = alg}) ∘ i₂ ≈ idC
+          i₂∘! = ⊤-id (F-Coalgebras (delayF Y) [ ! {A = alg} ∘ record { f = i₂ ; commutes = {!  inject₂ !} } ])
+          identityʳ' : extend ∘ now X ≈ f
+          identityʳ' = begin 
+            (F-Coalgebra-Morphism.f (! {A = alg}) ∘ i₁ {B = D₀ Y}) ∘ (out⁻¹ X ∘ i₁) ≈⟨ introˡ (_≅_.isoˡ (out-≅ Y)) ⟩ 
+            (out⁻¹ Y ∘ out Y) ∘ (F-Coalgebra-Morphism.f (! {A = alg}) ∘ i₁ {B = D₀ Y}) ∘ (out⁻¹ X ∘ i₁) ≈⟨ pullʳ (pullˡ (pullˡ (F-Coalgebra-Morphism.commutes (! {A = alg})))) ⟩
+            out⁻¹ Y ∘ (((idC +₁ (F-Coalgebra-Morphism.f !)) ∘ F-Coalgebra.α alg) ∘ i₁) ∘ out⁻¹ X ∘ i₁ ≈⟨ refl⟩∘⟨ (pullʳ (pullʳ (pullʳ (pullʳ (pullʳ (pullʳ +₁∘i₁)))))) ⟩∘⟨refl ⟩
+            out⁻¹ Y ∘ ((idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ (idC +₁ M._≅_.to +-assoc ∘ M._≅_.to +-comm) ∘ M._≅_.to +-assoc ∘ (out Y ∘ f +₁ idC) ∘ M._≅_.to +-assoc ∘ (i₁ ∘ out X)) ∘ out⁻¹ X ∘ i₁ ≈⟨ refl⟩∘⟨ (pullʳ (pullʳ (pullʳ (pullʳ (pullʳ (pullʳ (pullʳ (pullˡ (M._≅_.isoʳ (out-≅ X)))))))))) ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ (idC +₁ M._≅_.to +-assoc ∘ M._≅_.to +-comm) ∘ M._≅_.to +-assoc ∘ (out Y ∘ f +₁ idC) ∘ M._≅_.to +-assoc ∘ i₁ ∘ idC ∘ i₁ ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ identityˡ))))))) ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ (idC +₁ +-assocˡ ∘ M._≅_.to +-comm) ∘ M._≅_.to +-assoc ∘ (out Y ∘ f +₁ idC) ∘ M._≅_.to +-assoc ∘ i₁ ∘ i₁ ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ (+-assocˡ∘i₁∘i₁)))))) ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ (idC +₁ +-assocˡ ∘ M._≅_.to +-comm) ∘ M._≅_.to +-assoc ∘ (out Y ∘ f +₁ idC) ∘ i₁ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ +₁∘i₁ ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ (idC +₁ +-assocˡ ∘ M._≅_.to +-comm) ∘ M._≅_.to +-assoc ∘ (i₁ ∘ (out Y ∘ f)) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ inject₁ ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ (idC +₁ +-assocˡ ∘ M._≅_.to +-comm) ∘ [ i₁ , i₂ ∘ i₁ ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ ∘[] ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ [ (idC +₁ +-assocˡ ∘ M._≅_.to +-comm) ∘ i₁ , (idC +₁ +-assocˡ ∘ M._≅_.to +-comm) ∘ i₂ ∘ i₁ ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ [ i₁ ∘ idC , (i₂ ∘ (+-assocˡ ∘ M._≅_.to +-comm)) ∘ i₁ ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ([]-cong₂ identityʳ (pullʳ (pullʳ inject₁))) ⟩∘⟨refl ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ [ i₁ , i₂ ∘ (+-assocˡ ∘ i₂) ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ([]-cong₂ refl (refl⟩∘⟨ inject₂)) ⟩∘⟨refl ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ (idC +₁ idC +₁ [ idC , idC ]) ∘ [ i₁ , i₂ ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ ∘[] ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ [ (idC +₁ idC +₁ [ idC , idC ]) ∘ i₁ , (idC +₁ idC +₁ [ idC , idC ]) ∘ i₂ ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ []-cong₂ +₁∘i₁ (pullˡ +₁∘i₂) ⟩∘⟨refl ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ [ i₁ ∘ idC , (i₂ ∘ (idC +₁ [ idC , idC ])) ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ []-cong₂ identityʳ (pullˡ (pullʳ +₁∘i₂)) ⟩∘⟨refl ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ [ i₁ , (i₂ ∘ (i₂ ∘ [ idC , idC ])) ∘ i₂ ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ []-cong₂ refl (pullʳ (pullʳ inject₂)) ⟩∘⟨refl ⟩
+            out⁻¹ Y ∘ (idC +₁ F-Coalgebra-Morphism.f !) ∘ [ i₁ , i₂ ∘ (i₂ ∘ idC) ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ pullˡ ∘[] ⟩
+            out⁻¹ Y ∘ [ (idC +₁ F-Coalgebra-Morphism.f !) ∘ i₁ , (idC +₁ F-Coalgebra-Morphism.f !) ∘ i₂ ∘ (i₂ ∘ idC) ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ (([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl) ⟩
+            out⁻¹ Y ∘ [ i₁ ∘ idC , (i₂ ∘ F-Coalgebra-Morphism.f !) ∘ (i₂ ∘ idC) ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ (([]-cong₂ identityʳ (refl⟩∘⟨ identityʳ ○ assoc ○ refl⟩∘⟨ ⊤-id {!   !})) ⟩∘⟨refl) ⟩
+            out⁻¹ Y ∘ [ i₁ , i₂ ∘ idC ] ∘ (out Y ∘ f) ≈⟨ refl⟩∘⟨ (([]-cong₂ refl {!   !}) ⟩∘⟨refl) ⟩
+            {!   !} ≈⟨ {!   !} ⟩
+            {!   !} ≈⟨ {!   !} ⟩
+            f ∎
+
+
+
+          -- +-assocˡ∘i₁ = begin +-assocˡ ∘ i₁ ≈⟨ inject₁ ⟩ [ i₁ , i₂ ∘ i₁ ] ∎
+          -- +-assocˡ∘i₁∘i₁ = begin +-assocˡ ∘ i₁ ∘ i₁ ≈⟨ inject₁ ⟩ [ i₁ , i₂ ∘ i₁ ] ∎
+            -- given: out ∘ ! ≈ (! +₁ idC) ∘ long
+```
+
+Old definitions:
+
+```agda
+
   record DelayFunctor : Set (o ⊔ ℓ ⊔ e) where
     field
       D : Endofunctor C
@@ -59,7 +186,7 @@ First I postulate the Functor *D*, maybe I should derive it...
     field
       now : ∀ {X} → X ⇒ D₀ X
       later : ∀ {X} → D₀ X ⇒ D₀ X
-      isIso : ∀ {X} → IsIso [ now {X} , later {X} ]
+      isIso : ∀ {X} → IsIso ([ now {X} , later {X} ])
 
     out : ∀ {X} → D₀ X ⇒ X + D₀ X
     out {X} = IsIso.inv (isIso {X})
@@ -67,11 +194,7 @@ First I postulate the Functor *D*, maybe I should derive it...
     field
       coit : ∀ {X Y} → Y ⇒ X + Y → Y ⇒ D₀ X
       coit-law : ∀ {X Y} {f : Y ⇒ X + Y} → out ∘ (coit f) ≈ (idC +₁ (coit f)) ∘ f
-```
 
-Now let's define the monad:
-
-```agda
   record DelayMonad : Set (o ⊔ ℓ ⊔ e) where
     field
       D : DelayFunctor