From fceee34cee2d28086222a01f99e0a54dbe245924 Mon Sep 17 00:00:00 2001 From: Danila Fedorin Date: Sun, 30 Jul 2023 19:54:38 -0700 Subject: [PATCH] Finish associativity proof --- Map.agda | 97 ++++++++++++++++++++++++++++++++++---------------------- 1 file changed, 59 insertions(+), 38 deletions(-) diff --git a/Map.agda b/Map.agda index 9a9250d..d8ad9af 100644 --- a/Map.agda +++ b/Map.agda @@ -345,6 +345,9 @@ module _ (_≈_ : B → B → Set 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 + 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 @@ -353,47 +356,65 @@ module _ (f : B → B → B) where with Expr-Provenance f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈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)) + (f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁)) ... | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂)) rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ = - (v₁ , (refl , ImplInsert.union-preserves-∈₂ f k∉km₂ v₁∈m₁)) + (v₁ , (refl , I.union-preserves-∈₂ k∉km₂ v₁∈m₁)) ... | (_ , (in₂ {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-∈₁ f u₂ v₂∈m₂ k∉km₁)) + (v₂ , (refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁)) - union-assoc₁ : ∀ (m₁ m₂ m₃ : Map) → subset (_≡_) (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃)) - union-assoc₁ m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) k v k,v∈m₁₂m₃ - with Expr-Provenance f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃) - ... | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ = - let (k∉ke₁ , k∉ke₂) = ImplInsert.∉-union-∉-either f {l₁ = l₁} {l₂ = l₂} k∉ke₁₂ - in (v₃ , (refl , ImplInsert.union-preserves-∈₂ f k∉ke₁ (ImplInsert.union-preserves-∈₂ f k∉ke₂ v₃∈e₃))) - ... | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ = - (v₂ , (refl , ImplInsert.union-preserves-∈₂ f k∉ke₁ (ImplInsert.union-preserves-∈₁ f u₂ v₂∈e₂ k∉ke₃))) - ... | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ = - (f v₂ v₃ , (refl , ImplInsert.union-preserves-∈₂ f k∉ke₁ (ImplInsert.union-combines f u₂ u₃ v₂∈e₂ v₃∈e₃))) - ... | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ = - (v₁ , (refl , ImplInsert.union-preserves-∈₁ f u₁ v₁∈e₁ (ImplInsert.union-preserves-∉ f k∉ke₂ k∉ke₃))) - ... | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ = - (f v₁ v₃ , (refl , ImplInsert.union-combines f u₁ (ImplInsert.union-preserves-Unique f l₂ l₃ u₃) v₁∈e₁ (ImplInsert.union-preserves-∈₂ f k∉ke₂ v₃∈e₃))) - ... | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ = - (f v₁ v₂ , (refl , ImplInsert.union-combines f u₁ (ImplInsert.union-preserves-Unique f l₂ l₃ u₃) v₁∈e₁ (ImplInsert.union-preserves-∈₁ f u₂ v₂∈e₂ k∉ke₃))) - ... | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ = - (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , ImplInsert.union-combines f u₁ (ImplInsert.union-preserves-Unique f l₂ l₃ u₃) v₁∈e₁ (ImplInsert.union-combines f u₂ u₃ v₂∈e₂ v₃∈e₃))) + 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 + union-assoc₁ : subset (_≡_) (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃)) + union-assoc₁ k v k,v∈m₁₂m₃ + with Expr-Provenance f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃) + ... | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃)) + rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ = + let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂ + in (v₃ , (refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃))) + ... | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃)) + rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ = + (v₂ , (refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃))) + ... | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃)) + rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ = + (f v₂ v₃ , (refl , I.union-preserves-∈₂ k∉ke₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) + ... | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃)) + rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ = + (v₁ , (refl , I.union-preserves-∈₁ u₁ v₁∈e₁ (I.union-preserves-∉ k∉ke₂ k∉ke₃))) + ... | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃)) + rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ = + (f v₁ v₃ , (refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃))) + ... | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃) + rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ = + (f v₁ v₂ , (refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃))) + ... | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃)) + rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ = + (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) - union-assoc₂ : ∀ (m₁ m₂ m₃ : Map) → subset (_≡_) (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃) - union-assoc₂ m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) k v k,v∈m₁m₂₃ - with Expr-Provenance f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃) - ... | (_ , (in₂ k∉ke₁ (in₂ k∉e₂ (single {v₃} v₁∈e₃)) , v₃∈m₁m₂₃)) = {!!} - ... | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉e₃) , v₂∈m₁m₂₃)) = {!!} - ... | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₁∈e₃)) , v₂v₃∈m₁m₂₃)) = {!!} - ... | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₁₂ , v₁∈m₁m₂₃)) = {!!} - ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉e₂ (single {v₃} v₁∈e₃)) , v₁v₃∈m₁m₂₃)) = {!!} - ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉e₃) , v₁v₂∈m₁m₂₃)) = {!!} - ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₁∈e₃)) , v₁v₂v₃∈m₁m₂₃)) = {!!} + union-assoc₂ : subset (_≡_) (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃) + union-assoc₂ k v k,v∈m₁m₂₃ + with Expr-Provenance f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃) + ... | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃)) + rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ = + (v₃ , (refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃)) + ... | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃)) + rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ = + (v₂ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃)) + ... | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃)) + rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ = + (f v₂ v₃ , (refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃)) + ... | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃)) + rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ = + let (k∉ke₂ , k∉ke₃) = I.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃ + in (v₁ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃)) + ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃)) + rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ = + (f v₁ v₃ , (refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃)) + ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃)) + rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ = + (f v₁ v₂ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃)) + ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃)) + rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ = + (f (f v₁ v₂) v₃ , (sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))