Add more properties of update(s) and start work on provenance

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

Map.agda

@@ -283,6 +283,9 @@ private module ImplInsert (f : B → B → B) where
     ...                            | yes _ = (k' , f v v') ∷ xs
     ...                            | no _ = (k' , v') ∷ update k v xs
 
+    updates : List (A × B) → List (A × B) → List (A × B)
+    updates l₁ l₂ = foldr update l₂ l₁
+
     restrict : List (A × B) → List (A × B) → List (A × B)
     restrict l [] = []
     restrict l ((k' , v') ∷ xs) with ∈k-dec k' l
@@ -290,7 +293,7 @@ private module ImplInsert (f : B → B → B) where
     ...                           | no _ = restrict l xs
 
     intersect : List (A × B) → List (A × B) → List (A × B)
-    intersect l₁ l₂ = restrict l₁ (foldr update l₂ l₁)
+    intersect l₁ l₂ = restrict l₁ (updates l₁ l₂)
 
     update-keys : ∀ {k : A} {v : B} {l : List (A × B)} →
                   keys l ≡ keys (update k v l)
@@ -305,7 +308,7 @@ private module ImplInsert (f : B → B → B) where
     update-preserves-Unique {k} {v} {l} u rewrite update-keys {k} {v} {l} = u
 
     updates-preserve-Unique : ∀ {l₁ l₂ : List (A × B)} →
-                              Unique (keys l₂) → Unique (keys (foldr update l₂ l₁))
+                              Unique (keys l₂) → Unique (keys (updates l₁ l₂))
     updates-preserve-Unique {[]} u = u
     updates-preserve-Unique {(k , v) ∷ xs} u = update-preserves-Unique (updates-preserve-Unique {xs} u)
 
@@ -329,21 +332,64 @@ private module ImplInsert (f : B → B → B) where
                                  Unique (keys l₂) → Unique (keys (intersect l₁ l₂))
     intersect-preserves-Unique {l₁} u = restrict-preserves-Unique (updates-preserve-Unique {l₁} u)
 
-    restrict-needs-both : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
+    restrict-needs-both : ∀ {k : A} {l₁ l₂ : List (A × B)} →
-                          (k , v) ∈ restrict l₁ l₂ → (k ∈k l₁ × (k , v) ∈ l₂)
+                          k ∈k restrict l₁ l₂ → (k ∈k l₁ × k ∈k l₂)
-    restrict-needs-both {k} {v} {l₁} {[]} ()
+    restrict-needs-both {k} {l₁} {[]} ()
-    restrict-needs-both {k} {v} {l₁} {(k' , v') ∷ xs} k,v∈l₁l₂
+    restrict-needs-both {k} {l₁} {(k' , _) ∷ xs} k∈l₁l₂
-        with ∈k-dec k' l₁ | k,v∈l₁l₂
+        with ∈k-dec k' l₁ | k∈l₁l₂
-    ...   | yes k'∈kl₁    | here k,v≡k',v'
+    ...   | yes k'∈kl₁    | here k≡k'
-            rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v' =
+            rewrite k≡k' =
             (k'∈kl₁ , here refl)
-    ...   | yes _         | there k,v∈l₁xs =
+    ...   | yes _         | there k∈l₁xs =
-            let (k∈kl₁ , k,v∈xs) = restrict-needs-both k,v∈l₁xs
+            let (k∈kl₁ , k∈kxs) = restrict-needs-both k∈l₁xs
-            in (k∈kl₁ , there k,v∈xs)
+            in (k∈kl₁ , there k∈kxs)
-    ...   | no k'∉kl₁     | k,v∈l₁xs =
+    ...   | no k'∉kl₁     | k∈l₁xs =
-            let (k∈kl₁ , k,v∈xs) = restrict-needs-both k,v∈l₁xs
+            let (k∈kl₁ , k∈kxs) = restrict-needs-both k∈l₁xs
-            in (k∈kl₁ , there k,v∈xs)
+            in (k∈kl₁ , there k∈kxs)
 
+    update-preserves-∈ : ∀ {k k' : A} {v v' : B} {l : List (A × B)} →
+                         ¬ k ≡ k' → (k , v) ∈ l → (k , v) ∈ update k' v' l
+    update-preserves-∈ {k} {k'} {v} {v'} {(k'' , v'') ∷ xs} k≢k' (here k,v≡k'',v'')
+        rewrite cong proj₁ k,v≡k'',v'' rewrite cong proj₂ k,v≡k'',v''
+        with ≡-dec-A k' k''
+    ...   | yes k'≡k'' = absurd (k≢k' (sym k'≡k''))
+    ...   | no _ = here refl
+    update-preserves-∈ {k} {k'} {v} {v'} {(k'' , v'') ∷ xs} k≢k' (there k,v∈xs)
+        with ≡-dec-A k' k''
+    ...   | yes _ = there k,v∈xs
+    ...   | no _ = there (update-preserves-∈ k≢k' k,v∈xs)
+
+    updates-preserve-∈₂ : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
+                         ¬ k ∈k l₁ → (k , v) ∈ l₂ → (k , v) ∈ updates l₁ l₂
+    updates-preserve-∈₂ {k} {v} {[]} {l₂} _ k,v∈l₂ = k,v∈l₂
+    updates-preserve-∈₂ {k} {v} {(k' , v') ∷ xs} {l₂} k∉kl₁ k,v∈l₂ =
+        update-preserves-∈ (λ k≡k' → k∉kl₁ (here k≡k')) (updates-preserve-∈₂ (λ k∈kxs → k∉kl₁ (there k∈kxs)) k,v∈l₂)
+
+    update-combines : ∀ {k : A} {v v' : B} {l : List (A × B)} →
+                      Unique (keys l) → (k , v) ∈ l → (k , f v' v) ∈ update k v' l
+    update-combines {k} {v} {v'} {(k' , v'') ∷ xs} _ (here k,v=k',v'')
+        rewrite cong proj₁ k,v=k',v'' rewrite cong proj₂ k,v=k',v''
+        with ≡-dec-A k' k'
+    ...   | yes _ = here refl
+    ...   | no k'≢k' = absurd (k'≢k' refl)
+    update-combines {k} {v} {v'} {(k' , v'') ∷ xs} (push k'≢xs uxs) (there k,v∈xs)
+        with ≡-dec-A k k'
+    ...   | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v∈xs))
+    ...   | no _ = there (update-combines uxs k,v∈xs)
+
+    updates-combine : ∀ {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) ∈ updates l₁ l₂
+    updates-combine {k} {v₁} {v₂} {(k' , v') ∷ xs} {l₂} (push k'≢xs uxs₁) u₂ (here k,v₁≡k',v') k,v₂∈l₂
+        rewrite cong proj₁ k,v₁≡k',v' rewrite cong proj₂ k,v₁≡k',v' =
+        update-combines (updates-preserve-Unique {l₁ = xs} u₂) (updates-preserve-∈₂ (All¬-¬Any k'≢xs) k,v₂∈l₂)
+    updates-combine {k} {v₁} {v₂} {(k' , v') ∷ xs} {l₂} (push k'≢xs uxs₁) u₂ (there k,v₁∈xs) k,v₂∈l₂ =
+        update-preserves-∈ k≢k' (updates-combine uxs₁ u₂ k,v₁∈xs k,v₂∈l₂)
+        where
+            k≢k' : ¬ k ≡ k'
+            k≢k' with ≡-dec-A k k'
+            ...    | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v₁∈xs))
+            ...    | no  k≢k' = k≢k'
 
 
 Map : Set (a ⊔ b)
@@ -361,6 +407,7 @@ Map-functional {m = (l , ul)} k,v∈m k,v'∈m = ListAB-functional ul k,v∈m k,
 data Expr : Set (a ⊔ b) where
     `_ : Map → Expr
     _∪_ : Expr → Expr → Expr
+    _∩_ : Expr → Expr → Expr
 
 module _ (f : B → B → B) where
     open ImplInsert f renaming
@@ -375,15 +422,29 @@ module _ (f : B → B → B) where
     intersect : Map → Map → Map
     intersect (kvs₁ , _) (kvs₂ , uks₂) = (intersect-impl kvs₁ kvs₂ , intersect-preserves-Unique {kvs₁} {kvs₂} uks₂)
 
+module _ (fUnion : B → B → B) (fIntersect : B → B → B) where
+    open ImplInsert fUnion using
+        ( union-combines
+        ; union-preserves-∈₁
+        ; union-preserves-∈₂
+        ; union-preserves-∉
+        )
+
+    open ImplInsert fIntersect using
+        ( restrict-needs-both
+        ; updates
+        )
+
     ⟦_⟧ : Expr -> Map
     ⟦ ` m ⟧ = m
-    ⟦ e₁ ∪ e₂ ⟧ = union ⟦ e₁ ⟧ ⟦ e₂ ⟧
+    ⟦ e₁ ∪ e₂ ⟧ = union fUnion ⟦ e₁ ⟧ ⟦ e₂ ⟧
+    ⟦ e₁ ∩ e₂ ⟧ = intersect fIntersect ⟦ e₁ ⟧ ⟦ e₂ ⟧
 
     data Provenance (k : A) : B → Expr → Set (a ⊔ b) where
         single : ∀ {v : B} {m : Map} → (k , v) ∈ m → Provenance k v (` m)
         in₁ : ∀ {v : B} {e₁ e₂ : Expr} → Provenance k v e₁ → ¬ k ∈k ⟦ e₂ ⟧ → Provenance k v (e₁ ∪ e₂)
         in₂ : ∀ {v : B} {e₁ e₂ : Expr} → ¬ k ∈k ⟦ e₁ ⟧ → Provenance k v e₂ → Provenance k v (e₁ ∪ e₂)
-        bothᵘ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (f v₁ v₂) (e₁ ∪ e₂)
+        bothᵘ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (fUnion v₁ v₂) (e₁ ∪ e₂)
 
     Expr-Provenance : ∀ (k : A) (e : Expr) → k ∈k ⟦ e ⟧ → Σ B (λ v → (Provenance k v e × (k , v) ∈ ⟦ e ⟧))
     Expr-Provenance k (` m) k∈km = let (v , k,v∈m) = locate k∈km in (v , (single k,v∈m , k,v∈m))
@@ -392,7 +453,7 @@ module _ (f : 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  (f v₁ v₂ , (bothᵘ g₁ g₂ , union-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
+            in  (fUnion v₁ v₂ , (bothᵘ g₁ g₂ , union-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
     ...   | yes k∈ke₁ | no k∉ke₂ =
             let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
             in (v₁ , (in₁ g₁ k∉ke₂ , union-preserves-∈₁ (proj₂ ⟦ e₁ ⟧) k,v₁∈e₁ k∉ke₂))
@@ -400,6 +461,12 @@ module _ (f : B → B → B) where
             let (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
             in (v₂ , (in₂ k∉ke₁ g₂ , union-preserves-∈₂ k∉ke₁ k,v₂∈e₂))
     ...   | no k∉ke₁ | no k∉ke₂  = absurd (union-preserves-∉ k∉ke₁ k∉ke₂ k∈ke₁e₂)
+    Expr-Provenance k (e₁ ∩ e₂) k∈ke₁e₂
+        with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
+    ...   | yes k∈ke₁ | yes k∈ke₂ = {!!}
+    ...   | yes k∈ke₁ | no k∉ke₂ = {!!}
+    ...   | no k∉ke₁ | yes k∈ke₂ = {!!}
+    ...   | no k∉ke₁ | no k∉ke₂ = {!!}
 
 
 module _ (_≈_ : B → B → Set b) where
@@ -416,13 +483,18 @@ module _ (_≈_ : B → B → Set b) where
              (f : B → B → 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
+        -- for both -- it doesn't matter, since we don't use intersection in
+        -- these proofs.
+
         module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where
             union-idemp : ∀ (m : Map) → lift (union f m m) m
             union-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
                 where
                     mm-m-subset : subset (union f m m) m
                     mm-m-subset k v k,v∈mm
-                        with Expr-Provenance f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm)
+                        with Expr-Provenance f f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm)
                     ...   | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
                             rewrite Map-functional {m = m} v'∈m v''∈m
                             rewrite Map-functional {m = union f m m} k,v∈mm v'v''∈mm =
@@ -439,7 +511,7 @@ module _ (_≈_ : B → B → Set b) where
                 where
                     union-comm-subset : ∀ (m₁ m₂ : Map) → subset (union f m₁ m₂) (union f m₂ m₁)
                     union-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
-                        with Expr-Provenance f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
+                        with Expr-Provenance f f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
                     ...   | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
                             rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
                             (f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
@@ -456,7 +528,7 @@ module _ (_≈_ : B → B → Set b) where
                 where
                     union-assoc₁ : subset (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
                     union-assoc₁ k v k,v∈m₁₂m₃
-                        with Expr-Provenance f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
+                        with Expr-Provenance f f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
                     ...   | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃))
                             rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ =
                             let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂
@@ -482,7 +554,7 @@ module _ (_≈_ : B → B → Set b) where
 
                     union-assoc₂ : subset (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃)
                     union-assoc₂ k v k,v∈m₁m₂₃
-                        with Expr-Provenance f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
+                        with Expr-Provenance f f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
                     ...   | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃))
                             rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ =
                             (v₃ , (≈-refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃))