further work on Collections
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164
src/Collections.lagda
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164
src/Collections.lagda
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---
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title : "Collections: Collections represented as Lists"
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layout : page
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permalink : /Collections
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---
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This chapter presents operations on collections and a number of
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useful operations on them.
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## Imports
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\begin{code}
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_; _≢_; refl; sym; trans; cong)
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open Eq.≡-Reasoning
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open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_; _≤_; s≤s; z≤n)
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-- open import Data.Nat.Properties using
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-- (+-assoc; +-identityˡ; +-identityʳ; *-assoc; *-identityˡ; *-identityʳ)
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open import Relation.Nullary using (¬_)
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open import Data.Product using (_×_) renaming (_,_ to ⟨_,_⟩)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Data.Empty using (⊥; ⊥-elim)
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open import Isomorphism using (_≃_)
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open import Function using (_∘_)
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open import Level using (Level)
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open import Data.List using (List; []; _∷_; _++_; map; foldr; filter)
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open import Data.List.All using (All; []; _∷_)
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open import Data.List.Any using (Any; here; there)
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open import Data.Maybe using (Maybe; just; nothing)
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-- open import Data.List.Any.Membership.Propositional using (_∈_)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Relation.Nullary.Negation using (contraposition; ¬?)
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open import Relation.Nullary.Product using (_×-dec_)
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-- open import Relation.Binary using (IsEquivalence)
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\end{code}
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## Collections
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\begin{code}
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infix 0 _↔_
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_↔_ : Set → Set → Set
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A ↔ B = (A → B) × (B → A)
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module Collection (A : Set) (_≟_ : ∀ (x y : A) → Dec (x ≡ y)) where
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-- abstract
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Coll : Set → Set
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Coll A = List A
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[_] : A → Coll A
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[ x ] = x ∷ []
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infix 4 _∈_
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infix 4 _⊆_
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infixl 5 _∪_
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infixl 5 _\\_
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data _∈_ : A → Coll A → Set where
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here : ∀ {x xs} →
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----------
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x ∈ x ∷ xs
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there : ∀ {w x xs} →
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w ∈ xs →
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----------
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w ∈ x ∷ xs
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_⊆_ : Coll A → Coll A → Set
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xs ⊆ ys = ∀ {w} → w ∈ xs → w ∈ ys
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_∪_ : Coll A → Coll A → Coll A
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_∪_ = _++_
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_\\_ : Coll A → A → Coll A
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xs \\ x = filter (¬? ∘ (_≟ x)) xs
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lemma₅ : ∀ {w x xs} → w ∈ xs → w ≢ x → w ∈ xs \\ x
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lemma₅ {w} {x} here w≢ with w ≟ x
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... | yes refl = ⊥-elim (w≢ refl)
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... | no _ = here
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lemma₅ {_} {x} {y ∷ _} (there w∈) w≢ with y ≟ x
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... | yes refl = lemma₅ w∈ w≢
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... | no _ = there (lemma₅ w∈ w≢)
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lemma₆ : ∀ {x : A} {xs ys : Coll A} → xs \\ x ⊆ ys ↔ xs ⊆ x ∷ ys
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lemma₆ = ⟨ forward , backward ⟩
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where
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forward : ∀ {x xs ys} → xs \\ x ⊆ ys → xs ⊆ x ∷ ys
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forward {x} ⊆ys {w} w∈ with w ≟ x
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... | yes refl = here
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... | no w≢ = there (⊆ys (lemma₅ w∈ w≢))
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backward = {!!}
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⊆-refl : ∀ {xs} → xs ⊆ xs
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⊆-refl ∈xs = ∈xs
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⊆-trans : ∀ {xs ys zs} → xs ⊆ ys → ys ⊆ zs → xs ⊆ zs
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⊆-trans xs⊆ ys⊆ = ys⊆ ∘ xs⊆
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lemma₁ : ∀ {w xs} → w ∈ xs ↔ [ w ] ⊆ xs
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lemma₁ = ⟨ forward , backward ⟩
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where
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forward : ∀ {w xs} → w ∈ xs → [ w ] ⊆ xs
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forward ∈xs here = ∈xs
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forward ∈xs (there ())
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backward : ∀ {w xs} → [ w ] ⊆ xs → w ∈ xs
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backward ⊆xs = ⊆xs here
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lemma₂ : ∀ {xs ys} → xs ⊆ xs ∪ ys
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lemma₂ here = here
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lemma₂ (there ∈xs) = there (lemma₂ ∈xs)
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lemma₃ : ∀ {xs ys} → ys ⊆ xs ∪ ys
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lemma₃ {[]} ∈ys = ∈ys
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lemma₃ {x ∷ xs} ∈ys = there (lemma₃ {xs} ∈ys)
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lemma₄ : ∀ {w xs ys} → w ∈ xs ⊎ w ∈ ys ↔ w ∈ xs ∪ ys
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lemma₄ = ⟨ forward , backward ⟩
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where
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forward : ∀ {w xs ys} → w ∈ xs ⊎ w ∈ ys → w ∈ xs ∪ ys
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forward (inj₁ ∈xs) = lemma₂ ∈xs
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forward (inj₂ ∈ys) = lemma₃ ∈ys
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backward : ∀ {xs ys w} → w ∈ xs ∪ ys → w ∈ xs ⊎ w ∈ ys
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backward {[]} ∈ys = inj₂ ∈ys
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backward {x ∷ xs} here = inj₁ here
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backward {x ∷ xs} (there w∈) with backward {xs} w∈
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... | inj₁ ∈xs = inj₁ (there ∈xs)
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... | inj₂ ∈ys = inj₂ ∈ys
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\end{code}
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## Standard Library
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Definitions similar to those in this chapter can be found in the standard library.
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\begin{code}
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-- EDIT
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\end{code}
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The standard library version of `IsMonoid` differs from the
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one given here, in that it is also parameterised on an equivalence relation.
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## Unicode
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This chapter uses the following unicode.
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EDIT
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∷ U+2237 PROPORTION (\::)
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⊗ U+2297 CIRCLED TIMES (\otimes)
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∈ U+2208 ELEMENT OF (\in)
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∉ U+2209 NOT AN ELEMENT OF (\inn)
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162
src/extra/Collections-setoid.lagda
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162
src/extra/Collections-setoid.lagda
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---
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title : "Collections: Collections represented as Lists"
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layout : page
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permalink : /Collections
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---
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This chapter presents operations on collections and a number of
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useful operations on them.
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## Imports
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\begin{code}
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-- import Relation.Binary.PropositionalEquality as Eq
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-- open Eq using (_≡_; refl; sym; trans; cong)
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-- open Eq.≡-Reasoning
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open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_; _≤_; s≤s; z≤n)
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-- open import Data.Nat.Properties using
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-- (+-assoc; +-identityˡ; +-identityʳ; *-assoc; *-identityˡ; *-identityʳ)
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open import Relation.Nullary using (¬_)
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open import Data.Product using (_×_) renaming (_,_ to ⟨_,_⟩)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Isomorphism using (_≃_)
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open import Function using (_∘_)
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open import Level using (Level)
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open import Data.List using (List; []; _∷_; _++_; map; foldr; filter)
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open import Data.List.All using (All; []; _∷_)
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open import Data.List.Any using (Any; here; there)
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open import Data.Maybe using (Maybe; just; nothing)
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-- open import Data.List.Any.Membership.Propositional using (_∈_)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Relation.Nullary.Negation using (contraposition; ¬?)
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open import Relation.Nullary.Product using (_×-dec_)
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open import Relation.Binary using (IsEquivalence)
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\end{code}
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## Collections
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\begin{code}
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infix 0 _↔_
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_↔_ : Set → Set → Set
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A ↔ B = (A → B) × (B → A)
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module Collection
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(A : Set)
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(_≈_ : A → A → Set)
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(isEquivalence : IsEquivalence _≈_)
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where
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open IsEquivalence isEquivalence
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abstract
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Coll : Set → Set
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Coll A = List A
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\end{code}
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Collections support the same abbreviations as lists for writing
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collections with a small number of elements.
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\begin{code}
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[_] : A → Coll A
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[ x ] = x ∷ []
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\end{code}
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\begin{code}
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infix 4 _∈_
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infix 4 _⊆_
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infix 5 _∪_
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_∈_ : A → Coll A → Set
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w ∈ xs = Any (w ≈_) xs
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_⊆_ : Coll A → Coll A → Set
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xs ⊆ ys = ∀ {w} → w ∈ xs → w ∈ ys
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_∪_ : Coll A → Coll A → Coll A
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_∪_ = _++_
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preserves : ∀ {u v xs} → u ≈ v → u ∈ xs → v ∈ xs
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preserves u≈v (here u≈) = here (trans (sym u≈v) u≈)
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preserves u≈v (there u∈) = there (preserves u≈v u∈)
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⊆-refl : ∀ {xs} → xs ⊆ xs
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⊆-refl ∈xs = ∈xs
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⊆-trans : ∀ {xs ys zs} → xs ⊆ ys → ys ⊆ zs → xs ⊆ zs
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⊆-trans xs⊆ ys⊆ = ys⊆ ∘ xs⊆
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lemma₁ : ∀ {w xs} → w ∈ xs ↔ [ w ] ⊆ xs
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lemma₁ = ⟨ forward , backward ⟩
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where
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forward : ∀ {w xs} → w ∈ xs → [ w ] ⊆ xs
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forward ∈xs (here w≈) = preserves (sym w≈) ∈xs -- ∈xs
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forward ∈xs (there ())
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backward : ∀ {w xs} → [ w ] ⊆ xs → w ∈ xs
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backward ⊆xs = ⊆xs (here refl)
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lemma₂ : ∀ {xs ys} → xs ⊆ xs ∪ ys
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lemma₂ (here w≈) = here w≈
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lemma₂ (there ∈xs) = there (lemma₂ ∈xs)
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lemma₃ : ∀ {xs ys} → ys ⊆ xs ∪ ys
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lemma₃ {[]} ∈ys = ∈ys
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lemma₃ {x ∷ xs} ∈ys = there (lemma₃ {xs} ∈ys)
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lemma₄ : ∀ {w xs ys} → w ∈ xs ⊎ w ∈ ys ↔ w ∈ xs ∪ ys
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lemma₄ = ⟨ forward , backward ⟩
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where
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forward : ∀ {w xs ys} → w ∈ xs ⊎ w ∈ ys → w ∈ xs ∪ ys
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forward (inj₁ ∈xs) = lemma₂ ∈xs
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forward (inj₂ ∈ys) = lemma₃ ∈ys
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backward : ∀ {xs ys w} → w ∈ xs ∪ ys → w ∈ xs ⊎ w ∈ ys
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backward {[]} ∈ys = inj₂ ∈ys
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backward {x ∷ xs} (here w≈) = inj₁ (here w≈)
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backward {x ∷ xs} (there w∈) with backward {xs} w∈
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... | inj₁ ∈xs = inj₁ (there ∈xs)
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... | inj₂ ∈ys = inj₂ ∈ys
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\end{code}
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# Operations with decidable equivalence
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\begin{code}
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module DecCollection (_≟_ : ∀ (x y : A) → Dec (x ≈ y)) where
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abstract
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infix 5 _\\_
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_\\_ : Coll A → A → Coll A
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xs \\ x = filter (¬? ∘ (x ≟_)) xs
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\end{code}
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## Standard Library
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Definitions similar to those in this chapter can be found in the standard library.
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\begin{code}
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-- EDIT
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\end{code}
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The standard library version of `IsMonoid` differs from the
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one given here, in that it is also parameterised on an equivalence relation.
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## Unicode
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This chapter uses the following unicode.
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EDIT
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∷ U+2237 PROPORTION (\::)
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⊗ U+2297 CIRCLED TIMES (\otimes)
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∈ U+2208 ELEMENT OF (\in)
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∉ U+2209 NOT AN ELEMENT OF (\inn)
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19
src/extra/ModuleInfix.agda
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19
src/extra/ModuleInfix.agda
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@ -0,0 +1,19 @@
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module ModuleInfix where
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open import Data.List using (List; _∷_; [])
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open import Data.Bool using (Bool; true; false)
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module Sort(A : Set)(_≤_ : A → A → Bool)(_⊝_ : A → A → A)(zero : A) where
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infix 1 _≤_
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infix 2 _⊝_
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insert : A → List A → List A
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insert x [] = x ∷ []
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insert x (y ∷ ys) with zero ≤ (y ⊝ x)
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insert x (y ∷ ys) | true = x ∷ y ∷ ys
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insert x (y ∷ ys) | false = y ∷ insert x ys
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sort : List A → List A
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sort [] = []
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sort (x ∷ xs) = insert x (sort xs)
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