From a4eaa6269fc26710f542a10d235278bf022f8953 Mon Sep 17 00:00:00 2001
From: Danila Fedorin <danila.fedorin@gmail.com>
Date: Sun, 30 Jul 2023 20:11:02 -0700
Subject: [PATCH] Prove idempotence

---
 Map.agda | 22 +++++++++++++++++++---
 1 file changed, 19 insertions(+), 3 deletions(-)

diff --git a/Map.agda b/Map.agda
index d8ad9af..1079cbd 100644
--- a/Map.agda
+++ b/Map.agda
@@ -343,11 +343,26 @@ module _ (_≈_ : B → B → Set b) where
     lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
 
 module _ (f : B → B → B) where
-    module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≡ f b₂ b₁)
-             (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≡ f b₁ (f b₂ b₃)) where
+    module I = ImplInsert f
 
-        module I = ImplInsert f
+    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)
+                ...   | (_ , (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 =
+                        (v'' , (f-idemp v'' , v''∈m))
+                ...   | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = absurd (k∉km (∈-cong proj₁ v'∈m))
+                ...   | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = absurd (k∉km (∈-cong proj₁ v''∈m))
 
+                m-mm-subset : subset (_≡_) m (union f m m)
+                m-mm-subset k v k,v∈m = (f v v , (sym (f-idemp v) , I.union-combines u u k,v∈m k,v∈m))
+
+    module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≡ f b₂ b₁) where
         union-comm : ∀ (m₁ m₂ : Map) → lift (_≡_) (union f m₁ m₂) (union f m₂ m₁)
         union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁)
             where
@@ -364,6 +379,7 @@ module _ (f : B → B → B) where
                         rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
                         (v₂ , (refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁))
 
+    module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≡ f b₁ (f b₂ b₃))  where
         union-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (_≡_) (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
         union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂)
             where