Add a lattice instance for Map

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

Lattice.agda

@@ -42,7 +42,7 @@ record IsLattice {a} (A : Set a)
         absorb-⊓-⊔ : (x y : A) → (x ⊓ (x ⊔ y)) ≈ x
 
     open IsSemilattice joinSemilattice public
-    open IsSemilattice meetSemilattice public renaming
+    open IsSemilattice meetSemilattice public hiding (≈-equiv; ≈-refl; ≈-sym; ≈-trans) renaming
         ( ⊔-assoc to ⊓-assoc
         ; ⊔-comm to ⊓-comm
         ; ⊔-idemp to ⊓-idemp
@@ -350,3 +350,41 @@ module IsLatticeInstances where
             ; absorb-⊔-⊓ = absorb-⊔-⊓
             ; absorb-⊓-⊔ = absorb-⊓-⊔
             }
+
+    module ForMap {a} {A B : Set a}
+        (≡-dec-A : Decidable (_≡_ {a} {A}))
+        (_≈₂_ : B → B → Set a)
+        (_⊔₂_ : B → B → B)
+        (_⊓₂_ : B → B → B)
+        (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
+
+        open import Map A B ≡-dec-A
+        open IsLattice lB renaming
+            ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym
+            ; ⊔-idemp to ⊔₂-idemp; ⊓-idemp to ⊓₂-idemp
+            ; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
+            )
+
+        private
+            module MapJoin = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊔₂_ (IsLattice.joinSemilattice lB)
+            module MapMeet = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊓₂_ (IsLattice.meetSemilattice lB)
+
+            infix 4 _≈_
+            infixl 20 _⊔_
+
+            _≈_ : Map → Map → Set a
+            _≈_ = lift (_≈₂_)
+
+            _⊔_ : Map → Map → Map
+            m₁ ⊔ m₂ = union _⊔₂_ m₁ m₂
+
+            _⊓_ : Map → Map → Map
+            m₁ ⊓ m₂ = intersect _⊓₂_ m₁ m₂
+
+        MapIsLattice : IsLattice Map _≈_ _⊔_ _⊓_
+        MapIsLattice = record
+            { joinSemilattice = MapJoin.MapIsUnionSemilattice
+            ; meetSemilattice = MapMeet.MapIsIntersectSemilattice
+            ; absorb-⊔-⊓ = union-intersect-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂
+            ; absorb-⊓-⊔ = intersect-union-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂
+            }