Start working on the evaluation operation.

Proving monotonicity is the main hurdle here.

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

Lattice/FiniteMap.agda

@@ -69,9 +69,15 @@ module WithKeys (ks : List A) where
                 km₁≡ks
         )
 
+    _∈_ : A × B → FiniteMap → Set (a ⊔ℓ b)
+    _∈_ k,v (m₁ , _) = k,v ∈ˡ (proj₁ m₁)
+
     _∈k_ : A → FiniteMap → Set a
     _∈k_ k (m₁ , _) = k ∈ˡ (keysᵐ m₁)
 
+    locate : ∀ {k : A} {fm : FiniteMap} → k ∈k fm → Σ B (λ v → (k , v) ∈ fm)
+    locate {k} {fm = (m , _)} k∈kfm = locateᵐ {k} {m} k∈kfm
+
     _updating_via_ : FiniteMap → List A → (A → B) → FiniteMap
     _updating_via_ (m₁ , ksm₁≡ks) ks f =
         ( m₁ updatingᵐ ks via f

Lattice/FiniteValueMap.agda

@@ -33,7 +33,6 @@ open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A lB
     using
         ( subset-impl
         ; locate; forget
-        ; _∈_
         ; Map-functional
         ; Expr-Provenance
         ; Expr-Provenance-≡
@@ -103,7 +102,7 @@ module IterProdIsomorphism where
         _⊔ⁱᵖ_ {ks} = IP._⊔_ (length ks)
 
         _∈ᵐ_ : ∀ {ks : List A} → A × B → FiniteMap ks → Set
-        _∈ᵐ_ {ks} k,v fm = k,v ∈ proj₁ fm
+        _∈ᵐ_ {ks} = _∈_ ks
 
     -- The left inverse is: from (to x) = x
     from-to-inverseˡ : ∀ {ks : List A} (uks : Unique ks) →
@@ -156,7 +155,7 @@ module IterProdIsomorphism where
 
     private
         first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
-                           Σ B (λ v → (k , v) ∈ proj₁ fm)
+                           Σ B (λ v → (k , v) ∈ᵐ fm)
         first-key-in-map (((k , v) ∷ _ , _) , refl) = (v , here refl)
 
         from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →