Prove that variables in a program all come from the program's code

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

Lattice/Map.agda

@@ -553,11 +553,18 @@ open ImplInsert _⊔₂_ using
     ; union-preserves-∈₂
     ; union-preserves-∉
     ; union-preserves-∈k₁
+    ; union-preserves-∈k₂
     )
 
 ⊔-combines : ∀ {k : A} {v₁ v₂ : B} {m₁ m₂ : Map} → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → (k , v₁ ⊔₂ v₂) ∈ (m₁ ⊔ m₂)
 ⊔-combines {k} {v₁} {v₂} {kvs₁ , u₁} {kvs₂ , u₂} k,v₁∈m₁ k,v₂∈m₂ = union-combines u₁ u₂ k,v₁∈m₁ k,v₂∈m₂
 
+⊔-preserves-∈k₁ : ∀ {k : A} → {m₁ m₂ : Map} → k ∈k m₁ → k ∈k (m₁ ⊔ m₂)
+⊔-preserves-∈k₁ {k} {(l₁ , _)} {(l₂ , _)} k∈km₁ = union-preserves-∈k₁ {l₁ = l₁} {l₂ = l₂} k∈km₁
+
+⊔-preserves-∈k₂ : ∀ {k : A} → {m₁ m₂ : Map} → k ∈k m₂ → k ∈k (m₁ ⊔ m₂)
+⊔-preserves-∈k₂ {k} {(l₁ , _)} {(l₂ , _)} k∈km₁ = union-preserves-∈k₂ {l₁ = l₁} {l₂ = l₂} k∈km₁
+
 open ImplInsert _⊓₂_ using
     ( restrict-needs-both
     ; updates

Lattice/MapSet.agda

@@ -6,18 +6,28 @@ open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)
 module Lattice.MapSet {a : Level} (A : Set a) (≡-dec-A : Decidable (_≡_ {a} {A})) where
 
 open import Data.List using (List; map)
-open import Data.Product using (proj₁)
+open import Data.Product using (_,_; proj₁)
 open import Function using (_∘_)
 
 open import Lattice.Unit using (⊤; tt) renaming (_≈_ to _≈₂_; _⊔_ to _⊔₂_; _⊓_ to _⊓₂_; isLattice to ⊤-isLattice)
 import Lattice.Map
 
 private module UnitMap = Lattice.Map A ⊤ _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A ⊤-isLattice
-open UnitMap using (Map)
-open UnitMap using
-    ( _⊆_; _≈_; ≈-equiv; _⊔_; _⊓_; empty
-    ; isUnionSemilattice; isIntersectSemilattice; isLattice; lattice
-    ) public
+open UnitMap
+    using (Map; Expr; ⟦_⟧)
+    renaming
+        ( Expr-Provenance to Expr-Provenanceᵐ
+        )
+open UnitMap
+    using
+        ( _⊆_; _≈_; ≈-equiv; _⊔_; _⊓_; _∪_ ; _∩_ ; `_; empty; forget
+        ; isUnionSemilattice; isIntersectSemilattice; isLattice; lattice
+        ; Provenance
+        ; ⊔-preserves-∈k₁
+        ; ⊔-preserves-∈k₂
+        )
+    renaming (_∈k_ to _∈_) public
+open Provenance public
 
 MapSet : Set a
 MapSet = Map
@@ -27,3 +37,9 @@ to-List = map proj₁ ∘ proj₁
 
 insert : A → MapSet → MapSet
 insert k = UnitMap.insert k tt
+
+singleton : A → MapSet
+singleton k = UnitMap.insert k tt empty
+
+Expr-Provenance : ∀ (k : A) (e : Expr) → k ∈ ⟦ e ⟧ → Provenance k tt e
+Expr-Provenance k e k∈e = let (tt , (prov , _)) = Expr-Provenanceᵐ k e k∈e in prov