Add intermediate state for insertion proofs

Map.agda

@@ -12,15 +12,14 @@ import Data.List.Membership.Propositional as MemProp
 
 open import Relation.Nullary using (¬_)
 open import Data.Nat using (ℕ)
-open import Data.List using (List; []; _∷_; _++_)
+open import Data.List using (List; map; []; _∷_; _++_)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
 open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
 open import Data.Empty using (⊥)
 
 keys : List (A × B) → List A
-keys [] = []
+keys = map proj₁
-keys ((k , v) ∷ xs) = k ∷ keys xs
 
 data Unique {c} {C : Set c} : List C → Set c where
     empty : Unique []
@@ -63,6 +62,7 @@ private module ImplRelation (_≈_ : B → B → Set b) where
     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
+    open import Data.List using (map)
     open MemProp using (_∈_)
 
     private
@@ -117,6 +117,52 @@ private module ImplInsert (f : B → B → B) where
     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)
 
+    insert-preserves-other-keys : ∀ (k k' : A) (v v' : B) (l : List (A × B)) → ¬ k ≡ k' → (k , v) ∈ l → (k , v) ∈ insert k' v' l
+    insert-preserves-other-keys k k' v v' (x ∷ xs) k≢k' (here k,v=x) rewrite sym k,v=x with ≡-dec-A k' k
+    ...                                                                                  | yes k'≡k = absurd (k≢k' (sym k'≡k))
+    ...                                                                                  | no _ = here refl
+    insert-preserves-other-keys k k' v v' ((k'' , _) ∷ xs) k≢k' (there k,v∈xs) with ≡-dec-A k' k''
+    ...                                                                          | yes _ = there k,v∈xs
+    ...                                                                          | no _ = there (insert-preserves-other-keys k k' v v' xs k≢k' k,v∈xs)
+
+    merge-preserves-keys₁ : ∀ (k : A) (v : B) (l₁ l₂ : List (A × B)) → ¬ k ∈k l₁ → (k , v) ∈ l₂ → (k , v) ∈ merge l₁ l₂
+    merge-preserves-keys₁ k v [] l₂ _ k,v∈l₂ = k,v∈l₂
+    merge-preserves-keys₁ k v ((k' , v') ∷ xs₁) l₂ k∉kl₁ k,v∈l₂ =
+        let recursion = merge-preserves-keys₁ k v xs₁ l₂ (λ k∈xs₁ → k∉kl₁ (there k∈xs₁)) k,v∈l₂
+        in insert-preserves-other-keys k k' v v' _ (λ k≡k' → k∉kl₁ (here k≡k')) recursion
+
+    insert-preserves-other-key : ∀ (k : A) (v : B) (l : List (A × B)) → ¬ k ∈k l → (k , v) ∈ insert k v l
+    insert-preserves-other-key k v [] k∉kl = here refl
+    insert-preserves-other-key k v ((k' , v') ∷ xs) k∉kl with ≡-dec-A k k'
+    ...                                                    | yes k≡k' = absurd (k∉kl (here k≡k'))
+    ...                                                    | no _ = there (insert-preserves-other-key k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
+
+
+    ∈-cong : ∀ {c d} {C : Set c} {D : Set d} {c : C} {l : List C} → (f : C → D) → c ∈ l → f c ∈ map f l
+    ∈-cong f (here c≡c') = here (cong f c≡c')
+    ∈-cong f (there c∈xs) = there (∈-cong f c∈xs)
+
+    --   prove that ¬ k ∈k m → (k , v) ∈ insert k v m
+    merge-preserves-keys₂ : ∀ (k : A) (v : B) (l₁ l₂ : List (A × B)) → Unique (keys l₁) → (k , v) ∈ l₁ → ¬ k ∈k l₂ → (k , v) ∈ merge l₁ l₂
+    merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (here _) k∉kl₂ = {!!} -- hard!
+        -- where
+        --     rest : ∀ (l l' : List (A × B)) → All (λ k'' → ¬ k ≡ k'') (keys l) → ¬ k ∈k l' → ¬ k ∈k merge l l'
+        --     rest [] l' _ k∉kl' = k∉kl'
+        --     rest l [] (k≢l) _ = help 
+        --         where
+        --             help : ∀ (l : List (A × B)) → All (λ k'' → ¬ k ≡ k'') (keys l) → ¬ k ∈k l
+        --             help [] _ ()
+        --             help ((k'' , _) ∷ xs) (k≢k'' ∷ k≢xs) (here k≡k'') = k≢k'' k≡k''
+        --             help ((k'' , _) ∷ xs) (k≢k'' ∷ k≢xs) (there k∈kxs) = help xs k≢xs k∈kxs
+        --     -- rest (x@(k'' , _) ∷ xs) l' (k≢k'' ∷ k≢xs) k∉kl' with (≡-dec-A k'' = (rest xs l' k≢xs k∉kl')
+        --         -- where
+        --         --     help : ¬ k ∈k (merge (x ∷ xs) l') -- insert x (merge xs l')
+        --         --     help (here k≡k'') = {!!}
+        --         --     help (there k∈) = {!!}
+        --         -- let nested = (rest xs l' k≢xs k∉kl')
+
+
+
 Map : Set (a ⊔ b)
 Map = Σ (List (A × B)) (λ l → Unique (keys l))
 
@@ -154,6 +200,27 @@ module _ (f : B → B → B) where
     merge-provenance : ∀ (m₁ m₂ : Map) (k : A) → k ∈k merge m₁ m₂ → Σ (Provenance k m₁ m₂) MergeResult
     merge-provenance = {!!}
 
+    -- ------------------------------------------------------------------------
+    --
+    -- The following can be proven using plain properties of insert:
+    --
+    -- prove that ¬ k ∈k m₁ → (k , v) ∈ m₂ → (k , v) ∈ merge m₁ m₂ (done)
+    --   prove that k ≢ k' → (k , v) ∈ m → (k , v) ∈ insert k' v' m (done)
+    -- prove that (k , v) ∈ m₁ → ¬ k ∈k m₂ → (k , v) ∈ merge m₁ m₂ (stuck)
+    --   prove that ¬ k ∈k m → (k , v) ∈ insert k v m
+    --
+    -- ------------------------------------------------------------------------
+    --
+    -- The following relies on uniqueness, since inserts stops after the first encounter.
+    --
+    -- prove that (k , v) ∈ m₁ → (k , v') ∈ m₂ → (k, f v v') ∈ merge m₁ m₂
+    --
+    -- ------------------------------------------------------------------------
+    --
+    -- The following can probably be proven via keys.
+    --
+    -- prove that k ∉k m₁ → k ∉k m₂ → k ∉k merge m₁ m₂
+
 module _ (_≈_ : B → B → Set b) where
     open ImplRelation _≈_ renaming (subset to subset-impl)