From 461732244a01f47a9411f435e89d208ca333e0b3 Mon Sep 17 00:00:00 2001 From: Danila Fedorin Date: Wed, 26 Jul 2023 20:40:28 -0700 Subject: [PATCH] Finish all in/not-in proofs. Signed-off-by: Danila Fedorin --- Map.agda | 119 ++++++++++++++++++++++++++++++++++++++++--------------- 1 file changed, 87 insertions(+), 32 deletions(-) diff --git a/Map.agda b/Map.agda index 0304a0e..217565e 100644 --- a/Map.agda +++ b/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-preserves-other-key k v ((k' , v') ∷ xs) k∉kl + insert-fresh k v [] k∉kl = here refl + 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! - -- where - -- rest : ∀ (l l' : List (A × B)) → All (λ k'' → ¬ k ≡ k'') (keys l) → ¬ k ∈k l' → ¬ k ∈k merge l l' - -- rest [] l' _ k∉kl' = k∉kl' - -- 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') + merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (there k,v∈xs₁) k∉kl₂ = + insert-preserves-∈-right k k' v v' (merge xs₁ l₂) k≢k' k,v∈mxs₁l + where + k,v∈mxs₁l = merge-preserves-keys₂ k v xs₁ l₂ uxs₁ k,v∈xs₁ k∉kl₂ + 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 ∈k m → (k , v) ∈ insert k v m + -- 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 (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)