From d641f65615aed6c79a9f67ff45a1052e7ee039f1 Mon Sep 17 00:00:00 2001
From: Leon Vatthauer <leon.vatthauer@fau.de>
Date: Fri, 16 Feb 2024 14:33:29 +0100
Subject: [PATCH] minor

---
 Poset.agda      |  3 +-
 TotalOrder.agda | 98 +++++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 99 insertions(+), 2 deletions(-)
 create mode 100644 TotalOrder.agda

diff --git a/Poset.agda b/Poset.agda
index 0892d82..a826a07 100644
--- a/Poset.agda
+++ b/Poset.agda
@@ -75,7 +75,6 @@ Monad→Closure {𝑃 = 𝑃} 𝑀 = record
         open Poset 𝑃
         open Monad 𝑀
         open Functor F
-
 -- full proof
 Closure↔Monad : ∀ {𝑃 : Poset o ℓ₁ ℓ₂} → (Closure 𝑃 → Monad {o} {ℓ₂} {e} (Thin e 𝑃)) ∧ (Monad {o} {ℓ₂} {e} (Thin e 𝑃) → Closure 𝑃)
-Closure↔Monad = Closure→Monad , Monad→Closure
\ No newline at end of file
+Closure↔Monad = Closure→Monad , Monad→Closure
diff --git a/TotalOrder.agda b/TotalOrder.agda
new file mode 100644
index 0000000..10930dc
--- /dev/null
+++ b/TotalOrder.agda
@@ -0,0 +1,98 @@
+module TotalOrder where
+
+open import Level
+open import Relation.Binary using (Poset; TotalOrder)
+open import Categories.Monad
+open import Categories.Monad.Strong
+open import Categories.Category
+open import Categories.Category.Construction.Thin
+open import Categories.Category.Cartesian
+open import Categories.Category.BinaryProducts
+open import Categories.Object.Product
+open import Categories.Object.Terminal
+open import Categories.Functor renaming (id to Id)
+open import Categories.NaturalTransformation
+open import Categories.Category.Monoidal
+open import Data.Product renaming (_×_ to _∧_)
+open import Agda.Builtin.Unit
+open import Poset
+open import Data.Sum
+open import Relation.Nullary
+
+private
+    variable
+        o ℓ₁ ℓ₂ e : Level
+
+Closure→Cartesian : ∀ {𝑃 : Poset o ℓ₁ ℓ₂} → (Closure 𝑃) → Cartesian (Thin _ 𝑃)
+Closure→Cartesian {𝑃} Clo = record
+    { terminal = record
+        { ⊤ = _
+        ; ⊤-is-terminal = _
+        }
+    ; products = _
+    }
+    where
+        open Closure Clo
+
+module _ (𝑇 : TotalOrder o ℓ₁ ℓ₂) where 
+    -- Closure on total order
+    open TotalOrder 𝑇 renaming (poset to 𝑃)
+    TClosure : Set (o ⊔ ℓ₁ ⊔ ℓ₂)
+    TClosure = Closure 𝑃
+
+    postulate
+        em : ∀ {A : Set ℓ₂} → A ⊎ ¬ A
+
+    T-Product : ∀ (Clo : Closure 𝑃) (X Y : Carrier) → Product (Thin e 𝑃) X Y
+    T-Product Clo X Y with em {X ≤ Y} in eq 
+    ... | inj₁ x = {!   !}
+    ... | inj₂ x = {!   !}
+        where
+            open Closure Clo
+
+
+    -- TODO remove once certain that no prove needed
+    Thin-Monoidal : Monoidal (Thin e 𝑃)
+    Thin-Monoidal = monoidalHelper (Thin _ 𝑃) record
+        { ⊗ = record
+                { F₀ = λ p → {!   !}
+                ; F₁ = _
+                ; identity = _
+                ; homomorphism = _
+                ; F-resp-≈ = _
+                }
+        ; unit = _
+        ; unitorˡ = _
+        ; unitorʳ = _
+        ; associator = _
+        ; unitorˡ-commute = _
+        ; unitorʳ-commute = _
+        ; assoc-commute = _
+        ; triangle = _
+        ; pentagon = _
+        }
+    
+    module _ (c : Cartesian (Thin e 𝑃)) where 
+        AllMonadsStrong : Cartesian (Thin e 𝑃) → Monad (Thin e 𝑃) → (H : Monoidal (Thin e 𝑃)) → StrongMonad {C = (Thin e 𝑃)} H
+        AllMonadsStrong Cart 𝑀 Mon = record
+            { M = 𝑀
+            ; strength = record
+                { strengthen = ntHelper record
+                    { η = λ X → {!   !}
+                    ; commute = _
+                    }
+                ; identityˡ = _
+                ; η-comm = _
+                ; μ-η-comm = _
+                ; strength-assoc = _
+                }
+            }
+            where
+                open Cartesian Cart
+                open BinaryProducts products
+                open Category (Thin _ 𝑃)
+                open Equiv
+                open Monad 𝑀
+                open Functor F
+                t : ∀ X Y → F₀ X × F₀ Y ⇒ F₀ (X × Y)
+                t = _