Prove semilattice properties for intersect

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

Map.agda

@@ -433,6 +433,12 @@ private module ImplInsert (f : B → B → B) where
             ...    | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v₁∈xs))
             ...    | no  k≢k' = k≢k'
 
+    intersect-combines : ∀ {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)} →
+                      Unique (keys l₁) → Unique (keys l₂) →
+                      (k , v₁) ∈ l₁ → (k , v₂) ∈ l₂ → (k , f v₁ v₂) ∈ intersect l₁ l₂
+    intersect-combines u₁ u₂ k,v₁∈l₁ k,v₂∈l₂ =
+        restrict-preserves-∈₂ (∈-cong proj₁ k,v₁∈l₁) (updates-combine u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
+
 
 Map : Set (a ⊔ b)
 Map = Σ (List (A × B)) (λ l → Unique (keys l))
@@ -475,10 +481,9 @@ module _ (fUnion : B → B → B) (fIntersect : B → B → B) where
     open ImplInsert fIntersect using
         ( restrict-needs-both
         ; updates
-        ; updates-combine
         ; intersect-preserves-∉₁
         ; intersect-preserves-∉₂
-        ; restrict-preserves-∈₂
+        ; intersect-combines
         )
 
     ⟦_⟧ : Expr -> Map
@@ -513,7 +518,7 @@ module _ (fUnion : B → B → B) (fIntersect : B → B → B) where
     ...   | yes k∈ke₁ | yes k∈ke₂ =
             let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
                 (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
-            in (fIntersect v₁ v₂ , (bothⁱ g₁ g₂ , restrict-preserves-∈₂ k∈ke₁ (updates-combine (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂)))
+            in (fIntersect v₁ v₂ , (bothⁱ g₁ g₂ , intersect-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
     ...   | yes k∈ke₁ | no k∉ke₂ = absurd (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
     ...   | no k∉ke₁ | yes k∈ke₂ = absurd (intersect-preserves-∉₁ {l₂ = proj₁ ⟦ e₂ ⟧} k∉ke₁ k∈ke₁e₂)
     ...   | no k∉ke₁ | no k∉ke₂ = absurd (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
@@ -534,7 +539,7 @@ module _ (_≈_ : B → B → Set b) where
         module I = ImplInsert f
 
         -- The Provenance type requires both union and intersection functions,
-        -- but here we're working with union only. Just use the union function
+        -- but here we're working with one operation only. Just use the union function
         -- for both -- it doesn't matter, since we don't use intersection in
         -- these proofs.
 
@@ -627,3 +632,47 @@ module _ (_≈_ : B → B → Set b) where
                     ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
                             rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
                             (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))
+
+        module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where
+            intersect-idemp : ∀ (m : Map) → lift (intersect f m m) m
+            intersect-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
+                where
+                    mm-m-subset : subset (intersect f m m) m
+                    mm-m-subset k v k,v∈mm
+                        with Expr-Provenance f f k ((` m) ∩ (` m)) (∈-cong proj₁ k,v∈mm)
+                    ...   | (_ , (bothⁱ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
+                            rewrite Map-functional {m = m} v'∈m v''∈m
+                            rewrite Map-functional {m = intersect f m m} k,v∈mm v'v''∈mm =
+                            (v'' , (f-idemp v'' , v''∈m))
+
+                    m-mm-subset : subset m (intersect f m m)
+                    m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.intersect-combines u u k,v∈m k,v∈m))
+
+        module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where
+            intersect-comm : ∀ (m₁ m₂ : Map) → lift (intersect f m₁ m₂) (intersect f m₂ m₁)
+            intersect-comm m₁ m₂ = (intersect-comm-subset m₁ m₂ , intersect-comm-subset m₂ m₁)
+                where
+                    intersect-comm-subset : ∀ (m₁ m₂ : Map) → subset (intersect f m₁ m₂) (intersect f m₂ m₁)
+                    intersect-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
+                        with Expr-Provenance f f k ((` m₁) ∩ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
+                    ...   | (_ , (bothⁱ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
+                            rewrite Map-functional {m = intersect f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
+                            (f v₂ v₁ , (f-comm v₁ v₂ , I.intersect-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
+
+        module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃))  where
+            intersect-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (intersect f (intersect f m₁ m₂) m₃) (intersect f m₁ (intersect f m₂ m₃))
+            intersect-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (intersect-assoc₁ , intersect-assoc₂)
+                where
+                    intersect-assoc₁ : subset (intersect f (intersect f m₁ m₂) m₃) (intersect f m₁ (intersect f m₂ m₃))
+                    intersect-assoc₁ k v k,v∈m₁₂m₃
+                        with Expr-Provenance f f k (((` m₁) ∩ (` m₂)) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
+                    ...   | (_ , (bothⁱ (bothⁱ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
+                            rewrite Map-functional {m = intersect f (intersect f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
+                            (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.intersect-combines u₁ (I.intersect-preserves-Unique {l₂} {l₃} u₃) v₁∈e₁ (I.intersect-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
+
+                    intersect-assoc₂ : subset (intersect f m₁ (intersect f m₂ m₃)) (intersect f (intersect f m₁ m₂) m₃)
+                    intersect-assoc₂ k v k,v∈m₁m₂₃
+                        with Expr-Provenance f f k ((` m₁) ∩ ((` m₂) ∩ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
+                    ...   | (_ , (bothⁱ (single {v₁} v₁∈e₁) (bothⁱ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
+                            rewrite Map-functional {m = intersect f m₁ (intersect f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
+                            (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.intersect-combines (I.intersect-preserves-Unique {l₁} {l₂} u₂) u₃ (I.intersect-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))