Add a preorder instance for product

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Lattice.agda

@@ -85,11 +85,9 @@ record Lattice {a} (A : Set a) : Set (lsuc a) where
 
     open IsLattice isLattice public
 
-private module NatInstances where
+module PreorderInstances where
-    open Nat
+    module ForNat where
         open NatProps
-    open Eq
-    open Data.Sum
 
         NatPreorder : Preorder ℕ
         NatPreorder = record
@@ -101,6 +99,45 @@ private module NatInstances where
                 }
             }
 
+    module ForProd {a} {A B : Set a} {{ pA : Preorder A }} {{ pB : Preorder B }} where
+        open Eq
+
+        private
+            _≼_ : A × B → A × B → Set a
+            (a₁ ,  b₁) ≼ (a₂ , b₂) = Preorder._≼_ pA a₁ a₂ × Preorder._≼_ pB b₁ b₂
+
+            ispA = Preorder.isPreorder pA
+            ispB = Preorder.isPreorder pB
+
+            ≼-refl : {p : A × B} → p ≼ p
+            ≼-refl {(a , b)} = (IsPreorder.≼-refl ispA {a}, IsPreorder.≼-refl ispB {b})
+
+            ≼-trans : {p₁ p₂ p₃ : A × B} → p₁ ≼ p₂ → p₂ ≼ p₃ → p₁ ≼ p₃
+            ≼-trans (a₁≼a₂ , b₁≼b₂) (a₂≼a₃ , b₂≼b₃) =
+                ( IsPreorder.≼-trans ispA a₁≼a₂ a₂≼a₃
+                , IsPreorder.≼-trans ispB b₁≼b₂ b₂≼b₃
+                )
+
+            ≼-antisym : {p₁ p₂ : A × B} → p₁ ≼ p₂ → p₂ ≼ p₁ → p₁ ≡ p₂
+            ≼-antisym (a₁≼a₂ , b₁≼b₂) (a₂≼a₁ , b₂≼b₁) = cong₂ (_,_) (IsPreorder.≼-antisym ispA a₁≼a₂ a₂≼a₁) (IsPreorder.≼-antisym ispB b₁≼b₂ b₂≼b₁)
+
+        ProdPreorder : Preorder (A × B)
+        ProdPreorder = record
+            { _≼_ = _≼_
+            ; isPreorder = record
+                { ≼-refl = ≼-refl
+                ; ≼-trans = ≼-trans
+                ; ≼-antisym = ≼-antisym
+                }
+            }
+
+
+private module NatInstances where
+    open Nat
+    open NatProps
+    open Eq
+    open PreorderInstances.ForNat
+
     private
         max-bound₁ : {x y z : ℕ} → x ⊔ y ≡ z → x ≤ z
         max-bound₁ {x} {y} {z} x⊔y≡z rewrite sym x⊔y≡z rewrite ⊔-comm x y = m≤n⇒m≤o⊔n y (≤-refl)