Add a generic Map module and prove its induced equivalence relation

Lattice.agda

@@ -1,7 +1,7 @@
 module Lattice where
 
 import Data.Nat.Properties as NatProps
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; isEquivalence)
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym)
 open import Relation.Binary.Definitions
 open import Data.Nat as Nat using (ℕ; _≤_)
 open import Data.Product using (_×_; _,_)
@@ -68,6 +68,55 @@ record Lattice {a} (A : Set a) : Set (lsuc a) where
 
     open IsLattice isLattice public
 
+module IsEquivalenceInstances where
+    module ForMap {a b} (A : Set a) (B : Set b)
+        (≡-dec-A : Decidable (_≡_ {a} {A}))
+        (_≈₂_ : B → B → Set b)
+        (eB : IsEquivalence B _≈₂_) where
+
+        open import Map A B ≡-dec-A using (Map; lift; subset; insert)
+        open import Data.List using (_∷_; []) -- TODO: re-export these with nicer names from map
+        open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
+
+        open IsEquivalence eB renaming
+            ( ≈-refl to ≈₂-refl
+            ; ≈-sym to ≈₂-sym
+            ; ≈-trans to ≈₂-trans
+            )
+
+        private
+            _≈_ : Map → Map → Set (Agda.Primitive._⊔_ a b)
+            _≈_ = lift _≈₂_
+
+            _⊆_ : Map → Map → Set (Agda.Primitive._⊔_ a b)
+            _⊆_ = subset _≈₂_
+
+            ⊆-refl : {m : Map} →  m ⊆ m
+            ⊆-refl k v k,v∈m = (v , (≈₂-refl , k,v∈m))
+
+            ⊆-trans : {m₁ m₂ m₃ : Map} →  m₁ ⊆ m₂ → m₂ ⊆ m₃ → m₁ ⊆ m₃
+            ⊆-trans m₁⊆m₂ m₂⊆m₃ k v k,v∈m₁ =
+                let
+                    (v' , (v≈v' , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
+                    (v'' , (v'≈v'' , k,v''∈m₃)) = m₂⊆m₃ k v' k,v'∈m₂
+                in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃))
+
+            ≈-refl : {m : Map} → m ≈ m
+            ≈-refl {m} = (⊆-refl , ⊆-refl)
+
+            ≈-sym : {m₁ m₂ : Map} → m₁ ≈ m₂ → m₂ ≈ m₁
+            ≈-sym (m₁⊆m₂ , m₂⊆m₁) = (m₂⊆m₁ , m₁⊆m₂)
+
+            ≈-trans : {m₁ m₂ m₃ : Map} → m₁ ≈ m₂ → m₂ ≈ m₃ → m₁ ≈ m₃
+            ≈-trans (m₁⊆m₂ , m₂⊆m₁) (m₂⊆m₃ , m₃⊆m₂) = (⊆-trans m₁⊆m₂ m₂⊆m₃ , ⊆-trans m₃⊆m₂ m₂⊆m₁)
+
+            LiftEquivalence : IsEquivalence Map _≈_
+            LiftEquivalence = record
+                { ≈-refl = ≈-refl
+                ; ≈-sym = ≈-sym
+                ; ≈-trans = ≈-trans
+                }
+
 module IsSemilatticeInstances where
     module ForNat where
         open Nat

Map.agda

@@ -0,0 +1,42 @@
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
+open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Nullary using (Dec; yes; no)
+open import Agda.Primitive using (Level; _⊔_)
+
+module Map {a b : Level} (A : Set a) (B : Set b)
+    (≡-dec-A : Decidable (_≡_ {a} {A}))
+    where
+
+open import Data.Nat using (ℕ)
+open import Data.String using (String; _++_)
+open import Data.List using (List; []; _∷_)
+open import Data.List.Membership.Propositional using ()
+open import Data.Product using (_×_; _,_; Σ)
+open import Data.Unit using (⊤)
+open import Data.Empty using (⊥)
+
+Map : Set (a ⊔ b)
+Map = List (A × B)
+
+insert : (B → B → B) → A → B → Map → Map
+insert f k v [] = (k , v) ∷ []
+insert f k v (x@(k' , v') ∷ xs) with ≡-dec-A k k'
+...                             | yes _ = (k , f v v') ∷ xs
+...                             | no _ = x ∷ insert f k v xs
+
+foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> Map -> C
+foldr f b [] = b
+foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
+
+_∈_ : (A × B) → Map → Set (a ⊔ b)
+_∈_ p m = Data.List.Membership.Propositional._∈_ p m
+
+subset : ∀ (_≈_ : B → B → Set b) → Map → Map → Set (a ⊔ b)
+subset _≈_ m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
+
+lift : ∀ (_≈_ : B → B → Set b) → Map → Map → Set (a ⊔ b)
+lift _≈_ m₁ m₂ = (m₁ ⊆ m₂) × (m₂ ⊆ m₁)
+    where
+        _⊆_ : Map → Map → Set (a ⊔ b)
+        _⊆_ = subset _≈_