Finish all in/not-in proofs.

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

Map.agda

@@ -1,7 +1,7 @@
 open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.Core using (Rel)
-open import Relation.Nullary using (Dec; yes; no)
+open import Relation.Nullary using (Dec; yes; no; Reflects; ofʸ; ofⁿ)
 open import Agda.Primitive using (Level; _⊔_)
 
 module Map {a b : Level} (A : Set a) (B : Set b)
@@ -41,6 +41,10 @@ Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') =
         help {[]} _ = x'≢x ∷ []
         help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
 
+All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
+All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
+All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs
+
 absurd : ∀ {a} {A : Set a} →  ⊥ → A
 absurd ()
 
@@ -131,6 +135,10 @@ private module ImplInsert (f : B → B → B) where
     ∈-cong f (here c≡c') = here (cong f c≡c')
     ∈-cong f (there c∈xs) = there (∈-cong f c∈xs)
 
+    locate : ∀ (k : A) (l : List (A × B)) → k ∈k l → Σ B (λ v → (k , v) ∈ l)
+    locate k ((k' , v) ∷ xs) (here k≡k') rewrite k≡k' = (v , here refl)
+    locate k ((k' , v) ∷ xs) (there k∈kxs) = let (v , k,v∈xs) = locate k xs k∈kxs in (v , there k,v∈xs)
+
     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
@@ -145,53 +153,100 @@ private module ImplInsert (f : B → B → B) where
         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)) →
+    insert-preserves-∈-right : ∀ (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)
+    insert-preserves-∈-right 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)
+    insert-preserves-∈-right 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)
+    ...   | no _ = there (insert-preserves-∈-right k k' v v' xs k≢k' k,v∈xs)
+
+    insert-preserves-∈k-right : ∀ (k k' : A) (v' : B) (l : List (A × B)) →
+                                  ¬ k ≡ k' → k ∈k l → k ∈k insert k' v' l
+    insert-preserves-∈k-right k k' v' l k≢k' k∈kl =
+        let (v , k,v∈l) = locate k l k∈kl
+        in ∈-cong proj₁ (insert-preserves-∈-right k k' v v' l k≢k' k,v∈l)
+
+    insert-preserves-∉-right : ∀ (k k' : A) (v' : B) (l : List (A × B)) →
+                                  ¬ k ≡ k' → ¬ k ∈k l → ¬ k ∈k insert k' v' l
+    insert-preserves-∉-right k k' v' [] k≢k' k∉kl (here k≡k') = k≢k' k≡k'
+    insert-preserves-∉-right k k' v' [] k≢k' k∉kl (there ())
+    insert-preserves-∉-right k k' v' ((k'' , v'') ∷ xs) k≢k' k∉kl k∈kil
+        with ≡-dec-A k k''
+    ...   | yes k≡k'' = k∉kl (here k≡k'')
+    ...   | no k≢k'' with ≡-dec-A k' k'' | k∈kil
+    ...       | yes k'≡k'' | here k≡k'' = k≢k'' k≡k''
+    ...       | yes k'≡k'' | there k∈kxs = k∉kl (there k∈kxs)
+    ...       | no k'≢k''  | here k≡k'' = k∉kl (here k≡k'')
+    ...       | no k'≢k''  | there k∈kxs = insert-preserves-∉-right k k' v' xs k≢k'
+                                           (λ k∈kxs → k∉kl (there k∈kxs)) k∈kxs
+
+    merge-preserves-∉ : ∀ (k : A) (l₁ l₂ : List (A × B)) →
+                        ¬ k ∈k l₁ → ¬ k ∈k l₂ → ¬ k ∈k merge l₁ l₂
+    merge-preserves-∉ k [] l₂ _ k∉kl₂ = k∉kl₂
+    merge-preserves-∉ k ((k' , v') ∷ xs₁) l₂ k∉kl₁ k∉kl₂
+        with ≡-dec-A k k'
+    ...   | yes k≡k' = absurd (k∉kl₁ (here k≡k'))
+    ...   | no k≢k' = insert-preserves-∉-right k k' v' _ k≢k' (merge-preserves-∉ k xs₁ l₂ (λ k∈kxs₁ → k∉kl₁ (there k∈kxs₁)) k∉kl₂)
 
     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
+        in insert-preserves-∈-right 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)) →
+    insert-fresh : ∀ (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-fresh k v [] k∉kl = here refl
-    insert-preserves-other-key k v ((k' , v') ∷ xs) k∉kl
+    insert-fresh 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)))
+    ...   | no _ = there (insert-fresh k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
 
-
-    --   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!
+    merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (there k,v∈xs₁) k∉kl₂ =
-        -- where
+        insert-preserves-∈-right k k' v v' (merge xs₁ l₂) k≢k' k,v∈mxs₁l
-        --     rest : ∀ (l l' : List (A × B)) → All (λ k'' → ¬ k ≡ k'') (keys l) → ¬ k ∈k l' → ¬ k ∈k merge l l'
+        where
-        --     rest [] l' _ k∉kl' = k∉kl'
+            k,v∈mxs₁l = merge-preserves-keys₂ k v xs₁ l₂ uxs₁ k,v∈xs₁ k∉kl₂
-        --     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')
 
+            k≢k' : ¬ k ≡ k'
+            k≢k' with ≡-dec-A k k'
+            ...    | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v∈xs₁))
+            ...    | no  k≢k' = k≢k'
+    merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (here k,v≡k',v') k∉kl₂
+        rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v' =
+        insert-fresh k' v' _ (merge-preserves-∉ k' xs₁ l₂ (All¬-¬Any k'≢xs₁) k∉kl₂)
+
+    insert-combines : ∀ (k : A) (v v' : B) (l : List (A × B)) →
+                      Unique (keys l) → (k , v') ∈ l → (k , f v v') ∈ (insert k v l)
+    insert-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)
+    insert-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 k≢k' = there (insert-combines k v v' xs uxs k,v'∈xs)
+
+    merge-combines : forall (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) ∈ merge l₁ l₂
+    merge-combines k v₁ v₂ ((k' , v) ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) ul₂ (here k,v₁≡k',v) k,v₂∈l₂
+        rewrite cong proj₁ (sym (k,v₁≡k',v)) rewrite cong proj₂ (sym (k,v₁≡k',v)) =
+        insert-combines k v₁ v₂ _ (merge-preserves-Unique xs₁ l₂ ul₂) (merge-preserves-keys₁ k v₂ xs₁ l₂ (All¬-¬Any k'≢xs₁) k,v₂∈l₂)
+    merge-combines k v₁ v₂ ((k' , v) ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) ul₂ (there k,v₁∈xs₁) k,v₂∈l₂ =
+        insert-preserves-∈-right k k' (f v₁ v₂) v _ k≢k' (merge-combines k v₁ v₂ xs₁ l₂ uxs₁ ul₂ k,v₁∈xs₁ 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'≢xs₁ (∈-cong proj₁ k,v₁∈xs₁))
+            ...    | no  k≢k' = k≢k'
 
 
 Map : Set (a ⊔ b)
@@ -237,20 +292,20 @@ module _ (f : B → B → B) where
     --
     -- 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 , v) ∈ m₁ → ¬ k ∈k m₂ → (k , v) ∈ merge m₁ m₂ (done)
-    --   prove that ¬ k ∈k m → (k , v) ∈ insert k v m
+    --   prove that ¬ k ∈k m → (k , v) ∈ insert k v m (done)
     --
     -- ------------------------------------------------------------------------
     --
     -- 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
+    -- prove that (k , v) ∈ m₁ → (k , v') ∈ m₂ → (k, f v v') ∈ merge m₁ m₂ (done)
     --
     -- ------------------------------------------------------------------------
     --
     -- The following can probably be proven via keys.
     --
-    -- prove that k ∉k m₁ → k ∉k m₂ → k ∉k merge m₁ m₂
+    -- prove that k ∉k m₁ → k ∉k m₂ → k ∉k merge m₁ m₂ (done)
 
 module _ (_≈_ : B → B → Set b) where
     open ImplRelation _≈_ renaming (subset to subset-impl)