Reorganize a bit and start on provenance

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

Map.agda

@@ -40,26 +40,38 @@ Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x'
         help {[]} _ = x'≢x ∷ []
         help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
 
-Map : Set (a ⊔ b)
-Map = Σ (List (A × B)) (λ l → Unique (keys l))
-
-_∈_ : (A × B) → Map → Set (a ⊔ b)
-_∈_ p (kvs , _) = MemProp._∈_ p kvs
-
 absurd : ∀ {a} {A : Set a} →  ⊥ → A
 absurd ()
 
+private module _ where
+    open MemProp using (_∈_)
+
+    unique-not-in : ∀ {k : A} {v : B} {l : List (A × B)}  → ¬ (All (λ k' → ¬ k ≡ k') (keys l) × (k , v) ∈ l)
+    unique-not-in {l = (k' , _) ∷ xs} (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
+    unique-not-in {l = _ ∷ xs} (_ ∷ rest , there k,v'∈xs) = unique-not-in (rest , k,v'∈xs)
+
+    ListAB-functional : ∀ {k : A} {v v' : B} {l : List (A × B)} → Unique (keys l) → (k , v) ∈ l → (k , v') ∈ l → v ≡ v'
+    ListAB-functional _ (here k,v≡x) (here k,v'≡x) = cong proj₂ (trans k,v≡x (sym k,v'≡x))
+    ListAB-functional (push k≢xs _) (here k,v≡x) (there k,v'∈xs) rewrite sym k,v≡x = absurd (unique-not-in (k≢xs , k,v'∈xs))
+    ListAB-functional (push k≢xs _) (there k,v∈xs) (here k,v'≡x) rewrite sym k,v'≡x = absurd (unique-not-in (k≢xs , k,v∈xs))
+    ListAB-functional {l = _ ∷ xs } (push _ uxs) (there k,v∈xs) (there k,v'∈xs) = ListAB-functional uxs k,v∈xs k,v'∈xs
+
 private module ImplRelation (_≈_ : B → B → Set b) where
+    open MemProp using (_∈_)
+
     subset : List (A × B) → List (A × B) → Set (a ⊔ b)
-    subset m₁ m₂ = ∀ (k : A) (v : B) → MemProp._∈_ (k , v) m₁ → Σ B (λ v' → v ≈ v' × (MemProp._∈_ (k , v') m₂))
+    subset m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
 
 private module ImplInsert (f : B → B → B) where
-    _∈k_ : A → List (A × B) → Set a
+    open MemProp using (_∈_)
-    _∈k_ k m = MemProp._∈_ k (keys m)
 
-    foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> List (A × B) -> C
+    private
-    foldr f b [] = b
+        _∈k_ : A → List (A × B) → Set a
-    foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
+        _∈k_ k m = k ∈ (keys m)
+
+        foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> List (A × B) -> C
+        foldr f b [] = b
+        foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
 
     insert : A → B → List (A × B) → List (A × B)
     insert k v [] = (k , v) ∷ []
@@ -96,19 +108,31 @@ private module ImplInsert (f : B → B → B) where
                                                    witness (here k≡k') = k≢k' k≡k'
                                                    witness (there k∈kxs) = k∉kxs k∈kxs
 
-    insert-preserves-unique : ∀ (k : A) (v : B) (l : List (A × B)) → Unique (keys l) → Unique (keys (insert k v l))
+    insert-preserves-Unique : ∀ (k : A) (v : B) (l : List (A × B)) → Unique (keys l) → Unique (keys (insert k v l))
-    insert-preserves-unique k v l u with (∈k-dec k l)
+    insert-preserves-Unique k v l u with (∈k-dec k l)
     ...                               | yes k∈kl rewrite insert-keys-∈ k v l k∈kl = u
     ...                               | no k∉kl rewrite sym (insert-keys-∉ k v l k∉kl) = Unique-append k∉kl u
 
-    merge-preserves-unique : ∀ (l₁ l₂ : List (A × B)) → Unique (keys l₂) → Unique (keys (merge l₁ l₂))
+    merge-preserves-Unique : ∀ (l₁ l₂ : List (A × B)) → Unique (keys l₂) → Unique (keys (merge l₁ l₂))
-    merge-preserves-unique [] l₂ u₂ = u₂
+    merge-preserves-Unique [] l₂ u₂ = u₂
-    merge-preserves-unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-unique k₁ v₁ (merge xs₁ l₂) (merge-preserves-unique xs₁ l₂ u₂)
+    merge-preserves-Unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-Unique k₁ v₁ (merge xs₁ l₂) (merge-preserves-Unique xs₁ l₂ u₂)
 
-private
+Map : Set (a ⊔ b)
-    unique-not-in : ∀ {k : A} {v : B} {l : List (A × B)}  → ¬ (All (λ k' → ¬ k ≡ k') (keys l) × MemProp._∈_ (k , v) l)
+Map = Σ (List (A × B)) (λ l → Unique (keys l))
-    unique-not-in {l = (k' , _) ∷ xs} (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
+
-    unique-not-in {l = _ ∷ xs} (_ ∷ rest , there k,v'∈xs) = unique-not-in (rest , k,v'∈xs)
+_∈_ : (A × B) → Map → Set (a ⊔ b)
+_∈_ p (kvs , _) = MemProp._∈_ p kvs
+
+_∈k_ : A → Map → Set a
+_∈k_ k (kvs , _) = MemProp._∈_ k (keys kvs)
+
+Map-functional : ∀ {k : A} {v v' : B} {m : Map} → (k , v) ∈ m → (k , v') ∈ m → v ≡ v'
+Map-functional {m = (l , ul)} k,v∈m k,v'∈m = ListAB-functional ul k,v∈m k,v'∈m
+
+data Provenance (k : A) (m₁ m₂ : Map) : Set (a ⊔ b) where
+    both : (v₁ v₂ : B) → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → Provenance k m₁ m₂
+    in₁ : (v₁ : B) → (k , v₁) ∈ m₁ → ¬ k ∈k m₂ → Provenance k m₁ m₂
+    in₂ : (v₂ : B) → ¬ k ∈k m₁ → (k , v₂) ∈ m₂ → Provenance k m₁ m₂
 
 module _ (f : B → B → B) where
     open ImplInsert f renaming
@@ -117,11 +141,18 @@ module _ (f : B → B → B) where
         )
 
     insert : A → B → Map → Map
-    insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-unique k v kvs uks)
+    insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-Unique k v kvs uks)
 
     merge : Map → Map → Map
-    merge (kvs₁ , _) (kvs₂ , uks₂) = (merge-impl kvs₁ kvs₂ , merge-preserves-unique kvs₁ kvs₂ uks₂)
+    merge (kvs₁ , _) (kvs₂ , uks₂) = (merge-impl kvs₁ kvs₂ , merge-preserves-Unique kvs₁ kvs₂ uks₂)
 
+    MergeResult : {k : A} {m₁ m₂ : Map} → Provenance k m₁ m₂ → Set (a ⊔ b)
+    MergeResult {k} {m₁} {m₂} (both v₁ v₂ _ _) = (k , f v₁ v₂) ∈ merge m₁ m₂
+    MergeResult {k} {m₁} {m₂} (in₁ v₁ _ _) = (k , v₁) ∈ merge m₁ m₂
+    MergeResult {k} {m₁} {m₂} (in₂ v₂ _ _) = (k , v₂) ∈ merge m₁ m₂
+
+    merge-provenance : ∀ (m₁ m₂ : Map) (k : A) → k ∈k merge m₁ m₂ → Σ (Provenance k m₁ m₂) MergeResult
+    merge-provenance = {!!}
 
 module _ (_≈_ : B → B → Set b) where
     open ImplRelation _≈_ renaming (subset to subset-impl)
@@ -131,9 +162,3 @@ module _ (_≈_ : B → B → Set b) where
 
     lift : Map → Map → Set (a ⊔ b)
     lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
-
-Map-functional : ∀ {k : A} {v v' : B} {m : Map} → (k , v) ∈ m → (k , v') ∈ m → v ≡ v'
-Map-functional (here k,v≡x) (here k,v'≡x) = cong proj₂ (trans k,v≡x (sym k,v'≡x))
-Map-functional {m = (_ , push k≢xs _)} (here k,v≡x) (there k,v'∈xs) rewrite sym k,v≡x = absurd (unique-not-in (k≢xs , k,v'∈xs))
-Map-functional {m = (_ , push k≢xs _)} (there k,v∈xs) (here k,v'≡x) rewrite sym k,v'≡x = absurd (unique-not-in (k≢xs , k,v∈xs))
-Map-functional {m = (_ ∷ xs , push _ uxs)} (there k,v∈xs) (there k,v'∈xs) = Map-functional {m = (xs , uxs)} k,v∈xs k,v'∈xs