Make 'isLattice' for simple types be an instance

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

Lattice/Nat.agda

@@ -18,31 +18,32 @@ 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
 
-isMaxSemilattice : IsSemilattice ℕ _≡_ _⊔_
-isMaxSemilattice = record
-    { ≈-equiv = record
-        { ≈-refl = refl
-        ; ≈-sym = sym
-        ; ≈-trans = trans
+instance
+    isMaxSemilattice : IsSemilattice ℕ _≡_ _⊔_
+    isMaxSemilattice = record
+        { ≈-equiv = record
+            { ≈-refl = refl
+            ; ≈-sym = sym
+            ; ≈-trans = trans
+            }
+        ; ≈-⊔-cong = ≡-⊔-cong
+        ; ⊔-assoc = ⊔-assoc
+        ; ⊔-comm = ⊔-comm
+        ; ⊔-idemp = ⊔-idem
         }
-    ; ≈-⊔-cong = ≡-⊔-cong
-    ; ⊔-assoc = ⊔-assoc
-    ; ⊔-comm = ⊔-comm
-    ; ⊔-idemp = ⊔-idem
-    }
 
-isMinSemilattice : IsSemilattice ℕ _≡_ _⊓_
-isMinSemilattice = record
-    { ≈-equiv = record
-        { ≈-refl = refl
-        ; ≈-sym = sym
-        ; ≈-trans = trans
+    isMinSemilattice : IsSemilattice ℕ _≡_ _⊓_
+    isMinSemilattice = record
+        { ≈-equiv = record
+            { ≈-refl = refl
+            ; ≈-sym = sym
+            ; ≈-trans = trans
+            }
+        ; ≈-⊔-cong = ≡-⊓-cong
+        ; ⊔-assoc = ⊓-assoc
+        ; ⊔-comm = ⊓-comm
+        ; ⊔-idemp = ⊓-idem
         }
-    ; ≈-⊔-cong = ≡-⊓-cong
-    ; ⊔-assoc = ⊓-assoc
-    ; ⊔-comm = ⊓-comm
-    ; ⊔-idemp = ⊓-idem
-    }
 
 private
     max-bound₁ : {x y z : ℕ} → x ⊔ y ≡ z → x ≤ z
@@ -74,18 +75,19 @@ 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
 
-isLattice : IsLattice ℕ _≡_ _⊔_ _⊓_
-isLattice = record
-    { joinSemilattice = isMaxSemilattice
-    ; meetSemilattice = isMinSemilattice
-    ; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
-    ; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
-    }
+instance
+    isLattice : IsLattice ℕ _≡_ _⊔_ _⊓_
+    isLattice = record
+        { joinSemilattice = isMaxSemilattice
+        ; meetSemilattice = isMinSemilattice
+        ; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
+        ; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
+        }
 
-lattice : Lattice ℕ
-lattice = record
-    { _≈_ = _≡_
-    ; _⊔_ = _⊔_
-    ; _⊓_ = _⊓_
-    ; isLattice = isLattice
-    }
+    lattice : Lattice ℕ
+    lattice = record
+        { _≈_ = _≡_
+        ; _⊔_ = _⊔_
+        ; _⊓_ = _⊓_
+        ; isLattice = isLattice
+        }

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
+        }