Make 'isLattice' for simple types be an instance

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

Lattice/Unit.agda

@@ -45,14 +45,15 @@ tt ⊓ tt = tt
 ⊔-idemp : (x : ⊤) → (x ⊔ x) ≈ x
 ⊔-idemp tt = Eq.refl
 
-isJoinSemilattice : IsSemilattice ⊤ _≈_ _⊔_
-isJoinSemilattice = record
-    { ≈-equiv = ≈-equiv
-    ; ≈-⊔-cong = ≈-⊔-cong
-    ; ⊔-assoc = ⊔-assoc
-    ; ⊔-comm  = ⊔-comm
-    ; ⊔-idemp = ⊔-idemp
-    }
+instance
+    isJoinSemilattice : IsSemilattice ⊤ _≈_ _⊔_
+    isJoinSemilattice = record
+        { ≈-equiv = ≈-equiv
+        ; ≈-⊔-cong = ≈-⊔-cong
+        ; ⊔-assoc = ⊔-assoc
+        ; ⊔-comm  = ⊔-comm
+        ; ⊔-idemp = ⊔-idemp
+        }
 
 ≈-⊓-cong : ∀ {ab₁ ab₂ ab₃ ab₄} → ab₁ ≈ ab₂ → ab₃ ≈ ab₄ → (ab₁ ⊓ ab₃) ≈ (ab₂ ⊓ ab₄)
 ≈-⊓-cong {tt} {tt} {tt} {tt} _ _ = Eq.refl
@@ -66,36 +67,32 @@ isJoinSemilattice = record
 ⊓-idemp : (x : ⊤) → (x ⊓ x) ≈ x
 ⊓-idemp tt = Eq.refl
 
-isMeetSemilattice : IsSemilattice ⊤ _≈_ _⊓_
-isMeetSemilattice = record
-    { ≈-equiv = ≈-equiv
-    ; ≈-⊔-cong = ≈-⊓-cong
-    ; ⊔-assoc = ⊓-assoc
-    ; ⊔-comm  = ⊓-comm
-    ; ⊔-idemp = ⊓-idemp
-    }
+instance
+    isMeetSemilattice : IsSemilattice ⊤ _≈_ _⊓_
+    isMeetSemilattice = record
+        { ≈-equiv = ≈-equiv
+        ; ≈-⊔-cong = ≈-⊓-cong
+        ; ⊔-assoc = ⊓-assoc
+        ; ⊔-comm  = ⊓-comm
+        ; ⊔-idemp = ⊓-idemp
+        }
 
-absorb-⊔-⊓ : (x y : ⊤) → (x ⊔ (x ⊓ y)) ≈ x
-absorb-⊔-⊓ tt tt = Eq.refl
+instance
+    isLattice : IsLattice ⊤ _≈_ _⊔_ _⊓_
+    isLattice = record
+        { joinSemilattice = isJoinSemilattice
+        ; meetSemilattice = isMeetSemilattice
+        ; absorb-⊔-⊓ = λ { tt tt → Eq.refl }
+        ; absorb-⊓-⊔ = λ { tt tt → Eq.refl }
+        }
 
-absorb-⊓-⊔ : (x y : ⊤) → (x ⊓ (x ⊔ y)) ≈ x
-absorb-⊓-⊔ tt tt = Eq.refl
-
-isLattice : IsLattice ⊤ _≈_ _⊔_ _⊓_
-isLattice = record
-    { joinSemilattice = isJoinSemilattice
-    ; meetSemilattice = isMeetSemilattice
-    ; absorb-⊔-⊓ = absorb-⊔-⊓
-    ; absorb-⊓-⊔ = absorb-⊓-⊔
-    }
-
-lattice : Lattice ⊤
-lattice = record
-    { _≈_ = _≈_
-    ; _⊔_ = _⊔_
-    ; _⊓_ = _⊓_
-    ; isLattice = isLattice
-    }
+    lattice : Lattice ⊤
+    lattice = record
+        { _≈_ = _≈_
+        ; _⊔_ = _⊔_
+        ; _⊓_ = _⊓_
+        ; isLattice = isLattice
+        }
 
 open Chain _≈_ ≈-equiv (IsLattice._≺_ isLattice) (IsLattice.≺-cong isLattice)
 
@@ -107,25 +104,26 @@ private
     isLongest {tt} {tt} (step (tt⊔tt≈tt , tt̷≈tt) _ _) = ⊥-elim (tt̷≈tt refl)
     isLongest (done _) = z≤n
 
-fixedHeight : IsLattice.FixedHeight isLattice 0
-fixedHeight = record
-    { ⊥ = tt
-    ; ⊤ = tt
-    ; longestChain = longestChain
-    ; bounded = isLongest
-    }
+instance
+    fixedHeight : IsLattice.FixedHeight isLattice 0
+    fixedHeight = record
+        { ⊥ = tt
+        ; ⊤ = tt
+        ; longestChain = longestChain
+        ; bounded = isLongest
+        }
 
-isFiniteHeightLattice : IsFiniteHeightLattice ⊤ 0 _≈_ _⊔_ _⊓_
-isFiniteHeightLattice = record
-    { isLattice = isLattice
-    ; fixedHeight = fixedHeight
-    }
+    isFiniteHeightLattice : IsFiniteHeightLattice ⊤ 0 _≈_ _⊔_ _⊓_
+    isFiniteHeightLattice = record
+        { isLattice = isLattice
+        ; fixedHeight = fixedHeight
+        }
 
-finiteHeightLattice : FiniteHeightLattice ⊤
-finiteHeightLattice = record
-    { height = 0
-    ; _≈_ = _≈_
-    ; _⊔_ = _⊔_
-    ; _⊓_ = _⊓_
-    ; isFiniteHeightLattice = isFiniteHeightLattice
-    }
+    finiteHeightLattice : FiniteHeightLattice ⊤
+    finiteHeightLattice = record
+        { height = 0
+        ; _≈_ = _≈_
+        ; _⊔_ = _⊔_
+        ; _⊓_ = _⊓_
+        ; isFiniteHeightLattice = isFiniteHeightLattice
+        }