Use an expression-based provenance to make enumerating cases easier

This should come in handy for the associativity proof.

Map.agda

@@ -265,6 +265,10 @@ data Provenance (k : A) (m₁ m₂ : Map) : Set (a ⊔ b) where
     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
 
+data Expr : Set (a ⊔ b) where
+    `_ : Map → Expr
+    _∪_ : Expr → Expr → Expr
+
 module _ (f : B → B → B) where
     open ImplInsert f renaming
         ( insert to insert-impl
@@ -277,31 +281,32 @@ module _ (f : B → B → B) where
     union : Map → Map → Map
     union (kvs₁ , _) (kvs₂ , uks₂) = (union-impl kvs₁ kvs₂ , union-preserves-Unique kvs₁ kvs₂ uks₂)
 
-    MergeResult : {k : A} {m₁ m₂ : Map} → Provenance k m₁ m₂ → Set (a ⊔ b)
+    ⟦_⟧ : Expr -> Map
-    MergeResult {k} {m₁} {m₂} (both v₁ v₂ _ _) = (k , f v₁ v₂) ∈ union m₁ m₂
+    ⟦ ` m ⟧ = m
-    MergeResult {k} {m₁} {m₂} (in₁ v₁ _ _) = (k , v₁) ∈ union m₁ m₂
+    ⟦ e₁ ∪ e₂ ⟧ = union ⟦ e₁ ⟧ ⟦ e₂ ⟧
-    MergeResult {k} {m₁} {m₂} (in₂ v₂ _ _) = (k , v₂) ∈ union m₁ m₂
+
+    data Magic (k : A) : B → Expr → Set (a ⊔ b) where
+        single : ∀ {v : B} {m : Map} → (k , v) ∈ m → Magic k v (` m)
+        in₁ᵘ : ∀ {v : B} {e₁ e₂ : Expr} → Magic k v e₁ → ¬ k ∈k ⟦ e₂ ⟧ → Magic k v (e₁ ∪ e₂)
+        in₂ᵘ : ∀ {v : B} {e₁ e₂ : Expr} → ¬ k ∈k ⟦ e₁ ⟧ → Magic k v e₂ → Magic k v (e₁ ∪ e₂)
+        bothᵘ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Magic k v₁ e₁ → Magic k v₂ e₂ → Magic k (f v₁ v₂) (e₁ ∪ e₂)
+
+    Expr-Magic : ∀ (k : A) (e : Expr) → k ∈k ⟦ e ⟧ → Σ B (λ v → (Magic k v e × (k , v) ∈ ⟦ e ⟧))
+    Expr-Magic k (` m) k∈km = let (v , k,v∈m) = locate k∈km in (v , (single k,v∈m , k,v∈m))
+    Expr-Magic k (e₁ ∪ e₂) k∈ke₁e₂
+        with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
+    ...   | yes k∈ke₁ | yes k∈ke₂ =
+            let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Magic k e₁ k∈ke₁
+                (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Magic k e₂ k∈ke₂
+            in  (f 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-Magic k e₁ k∈ke₁
+            in (v₁ , (in₁ᵘ g₁ k∉ke₂ , union-preserves-∈₂ (proj₂ ⟦ e₁ ⟧) k,v₁∈e₁ k∉ke₂))
+    ...   | no k∉ke₁ | yes k∈ke₂ =
+            let (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Magic k e₂ k∈ke₂
+            in (v₂ , (in₂ᵘ k∉ke₁ g₂ , union-preserves-∈₁ k∉ke₁ k,v₂∈e₂))
+    ...   | no k∉ke₁ | no k∉ke₂  = absurd (union-preserves-∉ k∉ke₁ k∉ke₂ k∈ke₁e₂)
 
-    union-provenance : ∀ (m₁ m₂ : Map) (k : A) → k ∈k union m₁ m₂ → Σ (Provenance k m₁ m₂) MergeResult
-    union-provenance m₁@(l₁ , u₁) m₂@(l₂ , u₂) k k∈km₁m₂
-        with ∈k-dec k l₁ | ∈k-dec k l₂
-    ...   | yes k∈kl₁ | yes k∈kl₂ =
-        let
-            (v₁ , k,v₁∈l₁) = locate k∈kl₁
-            (v₂ , k,v₂∈l₂) = locate k∈kl₂
-        in
-            (both v₁ v₂ k,v₁∈l₁ k,v₂∈l₂ , union-combines u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
-    ...   | yes k∈kl₁ | no k∉kl₂ =
-        let
-            (v₁ , k,v₁∈l₁) = locate k∈kl₁
-        in
-            (in₁ v₁ k,v₁∈l₁ k∉kl₂ , union-preserves-∈₂ u₁ k,v₁∈l₁ k∉kl₂)
-    ...   | no k∉kl₁ | yes k∈kl₂ =
-        let
-            (v₂ , k,v₂∈l₂) = locate k∈kl₂
-        in
-            (in₂ v₂ k∉kl₁ k,v₂∈l₂ , union-preserves-∈₁ k∉kl₁ k,v₂∈l₂)
-    ...   | no k∉kl₁ | no k∉kl₂  = absurd (union-preserves-∉ k∉kl₁ k∉kl₂ k∈km₁m₂)
 
 module _ (_≈_ : B → B → Set b) where
     open ImplRelation _≈_ renaming (subset to subset-impl)
@@ -319,14 +324,13 @@ module _ (f : B → B → 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 union-provenance f m₁ m₂ k (∈-cong proj₁ k,v∈m₁m₂)
+                    with Expr-Magic f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
-                ...   | (both v₁ v₂ v₁∈m₁ v₂∈m₂ , v₁v₂∈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 , ImplInsert.union-combines f u₂ u₁ v₂∈m₂ v₁∈m₁))
-
+                ...   | (_ , (in₁ᵘ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂))
-                ...   | (in₁ v₁ v₁∈m₁ k∉km₂ , v₁∈m₁m₂)
                         rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
                         (v₁ , (refl , ImplInsert.union-preserves-∈₁ f k∉km₂ v₁∈m₁))
-                ...   | (in₂ v₂ k∉km₁ v₂∈m₂ , v₂∈m₁m₂)
+                ...   | (_ , (in₂ᵘ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂))
                         rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
                         (v₂ , (refl , ImplInsert.union-preserves-∈₂ f u₂ v₂∈m₂ k∉km₁))