Show that lifted equality preserves equivalences

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

Lattice/Builder.agda

@@ -29,6 +29,22 @@ lift-≈-map : ∀ {a b} {A : Set a} {B : Set b} (f : A → B)
 lift-≈-map f _≈ᵃ_ _≈ᵇ_ ≈ᵃ→≈ᵇ (just a₁) (just a₂) (≈-just a₁≈a₂) = ≈-just (≈ᵃ→≈ᵇ a₁ a₂ a₁≈a₂)
 lift-≈-map f _≈ᵃ_ _≈ᵇ_ ≈ᵃ→≈ᵇ nothing nothing ≈-nothing = ≈-nothing
 
+instance
+    lift-≈-Equivalence : ∀ {a} {A : Set a} {_≈_ : A → A → Set a} {{isEquivalence : IsEquivalence A _≈_}} → IsEquivalence (Maybe A) (lift-≈ _≈_)
+    lift-≈-Equivalence {{isEquivalence}} = record
+        { ≈-trans =
+            (λ { (≈-just a₁≈a₂) (≈-just a₂≈a₃) → ≈-just (IsEquivalence.≈-trans isEquivalence a₁≈a₂ a₂≈a₃)
+               ; ≈-nothing ≈-nothing → ≈-nothing
+               })
+        ; ≈-sym =
+            (λ { (≈-just a₁≈a₂) → ≈-just (IsEquivalence.≈-sym isEquivalence a₁≈a₂)
+               ; ≈-nothing → ≈-nothing
+               })
+        ; ≈-refl = λ { {just a} → ≈-just (IsEquivalence.≈-refl isEquivalence)
+                     ; {nothing} → ≈-nothing
+                     }
+        }
+
 PartialAssoc : ∀ {a} {A : Set a} (_≈_ : A → A → Set a) (_⊗_ : A → A → Maybe A) → Set a
 PartialAssoc {a} {A} _≈_ _⊗_ = ∀ (x y z : A) →  lift-≈ _≈_ ((x ⊗ y) >>= (_⊗ z)) ((y ⊗ z) >>= (x ⊗_))