Make 'isLattice' for simple types be an instance

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

Lattice/Nat.agda

@@ -18,6 +18,7 @@ private
     ≡-⊓-cong : ∀ {a₁ a₂ a₃ a₄} → a₁ ≡ a₂ → a₃ ≡ a₄ → (a₁ ⊓ a₃) ≡ (a₂ ⊓ a₄)
     ≡-⊓-cong a₁≡a₂ a₃≡a₄ rewrite a₁≡a₂ rewrite a₃≡a₄ = refl
 
+instance
     isMaxSemilattice : IsSemilattice ℕ _≡_ _⊔_
     isMaxSemilattice = record
         { ≈-equiv = record
@@ -74,6 +75,7 @@ private
             helper : x ⊔ (x ⊓ y) ≤ x ⊔ x  → x ⊔ x ≡ x → x ⊔ (x ⊓ y) ≤ x
             helper x⊔x⊓y≤x⊔x x⊔x≡x rewrite x⊔x≡x = x⊔x⊓y≤x⊔x
 
+instance
     isLattice : IsLattice ℕ _≡_ _⊔_ _⊓_
     isLattice = record
         { joinSemilattice = isMaxSemilattice

Lattice/Unit.agda

@@ -45,6 +45,7 @@ tt ⊓ tt = tt
 ⊔-idemp : (x : ⊤) → (x ⊔ x) ≈ x
 ⊔-idemp tt = Eq.refl
 
+instance
     isJoinSemilattice : IsSemilattice ⊤ _≈_ _⊔_
     isJoinSemilattice = record
         { ≈-equiv = ≈-equiv
@@ -66,6 +67,7 @@ isJoinSemilattice = record
 ⊓-idemp : (x : ⊤) → (x ⊓ x) ≈ x
 ⊓-idemp tt = Eq.refl
 
+instance
     isMeetSemilattice : IsSemilattice ⊤ _≈_ _⊓_
     isMeetSemilattice = record
         { ≈-equiv = ≈-equiv
@@ -75,18 +77,13 @@ isMeetSemilattice = record
         ; ⊔-idemp = ⊓-idemp
         }
 
-absorb-⊔-⊓ : (x y : ⊤) → (x ⊔ (x ⊓ y)) ≈ x
-absorb-⊔-⊓ tt tt = Eq.refl
-
-absorb-⊓-⊔ : (x y : ⊤) → (x ⊓ (x ⊔ y)) ≈ x
-absorb-⊓-⊔ tt tt = Eq.refl
-
+instance
     isLattice : IsLattice ⊤ _≈_ _⊔_ _⊓_
     isLattice = record
         { joinSemilattice = isJoinSemilattice
         ; meetSemilattice = isMeetSemilattice
-    ; absorb-⊔-⊓ = absorb-⊔-⊓
-    ; absorb-⊓-⊔ = absorb-⊓-⊔
+        ; absorb-⊔-⊓ = λ { tt tt → Eq.refl }
+        ; absorb-⊓-⊔ = λ { tt tt → Eq.refl }
         }
 
     lattice : Lattice ⊤
@@ -107,6 +104,7 @@ private
     isLongest {tt} {tt} (step (tt⊔tt≈tt , tt̷≈tt) _ _) = ⊥-elim (tt̷≈tt refl)
     isLongest (done _) = z≤n
 
+instance
     fixedHeight : IsLattice.FixedHeight isLattice 0
     fixedHeight = record
         { ⊥ = tt