Clean up 'Map' to hide implementation details, extract code

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

Lattice/Map.agda

@@ -19,6 +19,7 @@ open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-ex
 open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
 open import Data.Empty using (⊥; ⊥-elim)
 open import Equivalence
+open import Utils using (Unique; push; empty; Unique-append; All¬-¬Any)
 
 open IsLattice lB using () renaming
     ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym; ≈-trans to ≈₂-trans
@@ -28,34 +29,10 @@ open IsLattice lB using () renaming
     ; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
     )
 
-module ImplKeys where
+private module ImplKeys where
     keys : List (A × B) → List A
     keys = map proj₁
 
-data Unique {c} {C : Set c} : List C → Set c where
-    empty : Unique []
-    push : ∀ {x : C} {xs : List C}
-        → All (λ x' → ¬ x ≡ x') xs
-        → Unique xs
-        → Unique (x ∷ xs)
-
-Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} →
-                ¬ MemProp._∈_ x xs → Unique xs →  Unique (xs ++ (x ∷ []))
-Unique-append {c} {C} {x} {[]} _ _ = push [] empty
-Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') =
-    push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
-    where
-        x'≢x : ¬ x' ≡ x
-        x'≢x x'≡x = x∉xs (here (sym x'≡x))
-
-        help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (x ∷ []))
-        help {[]} _ = x'≢x ∷ []
-        help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
-
-All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
-All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
-All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs
-
 private module _ where
     open MemProp using (_∈_)
     open ImplKeys
@@ -559,6 +536,8 @@ Expr-Provenance k (e₁ ∩ e₂) k∈ke₁e₂
 ...   | no k∉ke₁ | yes k∈ke₂ = ⊥-elim (intersect-preserves-∉₁ {l₂ = proj₁ ⟦ e₂ ⟧} k∉ke₁ k∈ke₁e₂)
 ...   | no k∉ke₁ | no k∉ke₂ = ⊥-elim (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
 
+module _ (≈₂-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈₂ b₂)) where
+    private module _ where
         data SubsetInfo (m₁ m₂ : Map) : Set (a ⊔ℓ b) where
             extra : (k : A) → k ∈k m₁ → ¬ k ∈k m₂ → SubsetInfo m₁ m₂
             mismatch : (k : A) (v₁ v₂ : B) → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → ¬ v₁ ≈₂ v₂ → SubsetInfo m₁ m₂
@@ -576,7 +555,6 @@ SubsetInfo-to-dec {m₁} {m₂} (mismatch k v₁ v₂ k,v₁∈m₁ k,v₂∈m
                 in v₁̷≈v₂ (subst (λ v'' → v₁ ≈₂ v'') (Map-functional {k} {v'} {v₂} {m₂} k,v'∈m₂ k,v₂∈m₂) v₁≈v')) -- for some reason, can't just use subst...
         SubsetInfo-to-dec (fine m₁⊆m₂) = yes m₁⊆m₂
 
-module _ (≈₂-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈₂ b₂)) where
         compute-SubsetInfo : ∀ m₁ m₂ → SubsetInfo m₁ m₂
         compute-SubsetInfo ([] , _) m₂ = fine (λ k v ())
         compute-SubsetInfo m₁@((k , v) ∷ xs₁ , push k≢xs₁ uxs₁) m₂@(l₂ , u₂)

Utils.agda

@@ -0,0 +1,32 @@
+module Utils where
+
+open import Data.List using (List; []; _∷_; _++_)
+open import Data.List.Membership.Propositional using (_∈_)
+open import Data.List.Relation.Unary.All using (All; []; _∷_)
+open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
+open import Relation.Binary.PropositionalEquality using (_≡_; sym)
+open import Relation.Nullary using (¬_)
+
+data Unique {c} {C : Set c} : List C → Set c where
+    empty : Unique []
+    push : ∀ {x : C} {xs : List C}
+        → All (λ x' → ¬ x ≡ x') xs
+        → Unique xs
+        → Unique (x ∷ xs)
+
+Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} →
+                ¬ x ∈ xs → Unique xs →  Unique (xs ++ (x ∷ []))
+Unique-append {c} {C} {x} {[]} _ _ = push [] empty
+Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') =
+    push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
+    where
+        x'≢x : ¬ x' ≡ x
+        x'≢x x'≡x = x∉xs (here (sym x'≡x))
+
+        help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (x ∷ []))
+        help {[]} _ = x'≢x ∷ []
+        help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
+
+All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
+All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
+All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs