Switch maps (and consequently most of the code) to using instances

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

Lattice/AboveBelow.agda

@@ -79,8 +79,9 @@ data _≈_ : AboveBelow → AboveBelow → Set a where
 ≈-dec [ x ] ⊥ = no λ ()
 ≈-dec [ x ] ⊤ = no λ ()
 
-≈-Decidable : IsDecidable _≈_
-≈-Decidable = record { R-dec = ≈-dec }
+instance
+    ≈-Decidable : IsDecidable _≈_
+    ≈-Decidable = record { R-dec = ≈-dec }
 
 -- Any object can be wrapped in an 'above below' to make it a lattice,
 -- since ⊤ and ⊥ are the largest and least elements, and the rest are left
@@ -175,14 +176,15 @@ module Plain (x : A) where
     ⊔-idemp ⊥ = ≈-⊥-⊥
     ⊔-idemp [ x ] rewrite x≈y⇒[x]⊔[y]≡[x] (≈₁-refl {x}) = ≈-refl
 
-    isJoinSemilattice : IsSemilattice AboveBelow _≈_ _⊔_
-    isJoinSemilattice = record
-        { ≈-equiv = ≈-equiv
-        ; ≈-⊔-cong = ≈-⊔-cong
-        ; ⊔-assoc = ⊔-assoc
-        ; ⊔-comm = ⊔-comm
-        ; ⊔-idemp = ⊔-idemp
-        }
+    instance
+        isJoinSemilattice : IsSemilattice AboveBelow _≈_ _⊔_
+        isJoinSemilattice = record
+            { ≈-equiv = ≈-equiv
+            ; ≈-⊔-cong = ≈-⊔-cong
+            ; ⊔-assoc = ⊔-assoc
+            ; ⊔-comm = ⊔-comm
+            ; ⊔-idemp = ⊔-idemp
+            }
 
     _⊓_ : AboveBelow → AboveBelow → AboveBelow
     ⊥ ⊓ x = ⊥
@@ -268,14 +270,15 @@ module Plain (x : A) where
     ⊓-idemp ⊤ = ≈-⊤-⊤
     ⊓-idemp [ x ] rewrite x≈y⇒[x]⊓[y]≡[x] (≈₁-refl {x}) = ≈-refl
 
-    isMeetSemilattice : IsSemilattice AboveBelow _≈_ _⊓_
-    isMeetSemilattice = record
-        { ≈-equiv = ≈-equiv
-        ; ≈-⊔-cong = ≈-⊓-cong
-        ; ⊔-assoc = ⊓-assoc
-        ; ⊔-comm = ⊓-comm
-        ; ⊔-idemp = ⊓-idemp
-        }
+    instance
+        isMeetSemilattice : IsSemilattice AboveBelow _≈_ _⊓_
+        isMeetSemilattice = record
+            { ≈-equiv = ≈-equiv
+            ; ≈-⊔-cong = ≈-⊓-cong
+            ; ⊔-assoc = ⊓-assoc
+            ; ⊔-comm = ⊓-comm
+            ; ⊔-idemp = ⊓-idemp
+            }
 
     absorb-⊔-⊓ : ∀ (ab₁ ab₂ : AboveBelow) → (ab₁ ⊔ (ab₁ ⊓ ab₂)) ≈ ab₁
     absorb-⊔-⊓ ⊥ ab₂ rewrite ⊥⊓x≡⊥ ab₂ = ≈-⊥-⊥
@@ -300,21 +303,22 @@ module Plain (x : A) where
     ...   | no x̷≈y rewrite x̷≈y⇒[x]⊔[y]≡⊤ x̷≈y rewrite x⊓⊤≡x [ x ] = ≈-refl
 
 
-    isLattice : IsLattice AboveBelow _≈_ _⊔_ _⊓_
-    isLattice = record
-        { joinSemilattice = isJoinSemilattice
-        ; meetSemilattice = isMeetSemilattice
-        ; absorb-⊔-⊓ = absorb-⊔-⊓
-        ; absorb-⊓-⊔ = absorb-⊓-⊔
-        }
+    instance
+        isLattice : IsLattice AboveBelow _≈_ _⊔_ _⊓_
+        isLattice = record
+            { joinSemilattice = isJoinSemilattice
+            ; meetSemilattice = isMeetSemilattice
+            ; absorb-⊔-⊓ = absorb-⊔-⊓
+            ; absorb-⊓-⊔ = absorb-⊓-⊔
+            }
 
-    lattice : Lattice AboveBelow
-    lattice = record
-        { _≈_ = _≈_
-        ; _⊔_ = _⊔_
-        ; _⊓_ = _⊓_
-        ; isLattice = isLattice
-        }
+        lattice : Lattice AboveBelow
+        lattice = record
+            { _≈_ = _≈_
+            ; _⊔_ = _⊔_
+            ; _⊓_ = _⊓_
+            ; isLattice = isLattice
+            }
 
     open IsLattice isLattice using (_≼_; _≺_; ⊔-Monotonicˡ; ⊔-Monotonicʳ) public
 
@@ -360,25 +364,26 @@ module Plain (x : A) where
     isLongest {⊥} (step {_} {[ x ]} _ (≈-lift _) (step [x]≺y y≈z c@(step _ _ _)))
         rewrite [x]≺y⇒y≡⊤ _ _ [x]≺y with ≈-⊤-⊤ ← y≈z = ⊥-elim (¬-Chain-⊤ c)
 
-    fixedHeight : IsLattice.FixedHeight isLattice 2
-    fixedHeight = record
-        { ⊥ = ⊥
-        ; ⊤ = ⊤
-        ; longestChain = longestChain
-        ; bounded = isLongest
-        }
+    instance
+        fixedHeight : IsLattice.FixedHeight isLattice 2
+        fixedHeight = record
+            { ⊥ = ⊥
+            ; ⊤ = ⊤
+            ; longestChain = longestChain
+            ; bounded = isLongest
+            }
 
-    isFiniteHeightLattice : IsFiniteHeightLattice AboveBelow 2 _≈_ _⊔_ _⊓_
-    isFiniteHeightLattice = record
-        { isLattice = isLattice
-        ; fixedHeight = fixedHeight
-        }
+        isFiniteHeightLattice : IsFiniteHeightLattice AboveBelow 2 _≈_ _⊔_ _⊓_
+        isFiniteHeightLattice = record
+            { isLattice = isLattice
+            ; fixedHeight = fixedHeight
+            }
 
-    finiteHeightLattice : FiniteHeightLattice AboveBelow
-    finiteHeightLattice = record
-        { height = 2
-        ; _≈_ = _≈_
-        ; _⊔_ = _⊔_
-        ; _⊓_ = _⊓_
-        ; isFiniteHeightLattice = isFiniteHeightLattice
-        }
+        finiteHeightLattice : FiniteHeightLattice AboveBelow
+        finiteHeightLattice = record
+            { height = 2
+            ; _≈_ = _≈_
+            ; _⊔_ = _⊔_
+            ; _⊓_ = _⊓_
+            ; isFiniteHeightLattice = isFiniteHeightLattice
+            }