Tentative start on proving correctness

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

Language/Semantics.agda

@@ -2,6 +2,7 @@ module Language.Semantics where
 
 open import Language.Base
 
+open import Agda.Primitive using (lsuc)
 open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_)
 open import Data.Product using (_×_; _,_)
 open import Data.String using (String)
@@ -10,6 +11,9 @@ open import Data.Nat using (ℕ)
 open import Relation.Nullary using (¬_)
 open import Relation.Binary.PropositionalEquality using (_≡_)
 
+open import Lattice
+open import Utils using (_⇒_)
+
 data Value : Set where
     ↑ᶻ : ℤ → Value
 
@@ -58,3 +62,10 @@ data _,_⇒ˢ_ : Env → Stmt → Env → Set where
     ⇒ˢ-while-false : ∀ (ρ : Env) (e : Expr) (s : Stmt) →
         ρ , e ⇒ᵉ (↑ᶻ (+ 0)) →
         ρ , (while e repeat s) ⇒ˢ ρ
+
+record LatticeInterpretation {l} {L : Set l} {_≈_ : L → L → Set l}
+                             {_⊔_ : L → L → L} {_⊓_ : L → L → L}
+                             (isLattice : IsLattice L _≈_ _⊔_ _⊓_) : Set (lsuc l) where
+    field
+        ⟦_⟧ : L → Value → Set
+        ⟦⟧-respects-≈ : ∀ {l₁ l₂ : L} → l₁ ≈ l₂ → ⟦ l₁ ⟧ ⇒ ⟦ l₂ ⟧

Utils.agda

@@ -81,3 +81,7 @@ concat-∈ : ∀ {a} {A : Set a} {x : A} {l : List A} {ls : List (List A)} →
            x ∈ l → l ∈ ls → x ∈ foldr _++_ [] ls
 concat-∈ x∈l (here refl) = ListMemProp.∈-++⁺ˡ x∈l
 concat-∈ {ls = l' ∷ ls'} x∈l (there l∈ls') = ListMemProp.∈-++⁺ʳ l' (concat-∈ x∈l l∈ls')
+
+_⇒_ : ∀ {a p₁ p₂} {A : Set a} (P : A → Set p₁) (Q : A → Set p₂) →
+      Set (a ⊔ℓ p₁ ⊔ℓ p₂)
+_⇒_ P Q = ∀ a → P a → Q a