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Prove idempotence

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Danila Fedorin 2023-07-30 20:11:02 -07:00
parent fceee34cee
commit a4eaa6269f
1 changed files with 19 additions and 3 deletions
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Map.agda
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@ -343,11 +343,26 @@ module _ (_≈_ : B → B → Set b) where
lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁ lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
module _ (f : B → B → B) where 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 : Map) → lift (_≡_) (union f m₁ m₂) (union f m₂ m₁)
union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁) union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁)
where where
@ -364,6 +379,7 @@ module _ (f : B → B → B) where
rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ = rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
(v₂ , (refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁)) (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 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) union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂)
where where