------------------------------------------------------------------------ -- Lists where at least one element satisfies a given property ------------------------------------------------------------------------ module Data.List.Any where open import Data.Empty open import Data.Fin open import Data.Function open import Data.List as List using (List; []; _∷_) import Data.List.Equality as ListEq open import Data.Product as Prod using (∃; _×_; _,_) open import Relation.Nullary import Relation.Nullary.Decidable as Dec open import Relation.Unary using (Pred) renaming (_⊆_ to _⋐_) open import Relation.Binary open import Relation.Binary.PropositionalEquality as PropEq using (_≡_) -- Any P xs means that at least one element in xs satisfies P. data Any {A} (P : A → Set) : List A → Set where here : ∀ {x xs} (px : P x) → Any P (x ∷ xs) there : ∀ {x xs} (pxs : Any P xs) → Any P (x ∷ xs) -- Map. map : ∀ {A} {P Q : Pred A} → P ⋐ Q → Any P ⋐ Any Q map g (here px) = here (g px) map g (there pxs) = there (map g pxs) -- If the head does not satisfy the predicate, then the tail will. tail : ∀ {A x xs} {P : A → Set} → ¬ P x → Any P (x ∷ xs) → Any P xs tail ¬px (here px) = ⊥-elim (¬px px) tail ¬px (there pxs) = pxs -- Decides Any. any : ∀ {A} {P : A → Set} → (∀ x → Dec (P x)) → (xs : List A) → Dec (Any P xs) any p [] = no λ() any p (x ∷ xs) with p x any p (x ∷ xs) | yes px = yes (here px) any p (x ∷ xs) | no ¬px = Dec.map (there , tail ¬px) (any p xs) -- index x∈xs is the list position (zero-based) which x∈xs points to. index : ∀ {A} {P : A → Set} {xs} → Any P xs → Fin (List.length xs) index (here px) = zero index (there pxs) = suc (index pxs) ------------------------------------------------------------------------ -- List membership and some related definitions module Membership (S : Setoid) where private open module S = Setoid S using (_≈_) renaming (carrier to A) open module L = ListEq.Equality S using ([]; _∷_) renaming (_≈_ to _≋_) -- If a predicate P respects the underlying equality then Any P -- respects the list equality. lift-resp : ∀ {P} → P Respects _≈_ → Any P Respects _≋_ lift-resp resp [] () lift-resp resp (x≈y ∷ xs≈ys) (here px) = here (resp x≈y px) lift-resp resp (x≈y ∷ xs≈ys) (there pxs) = there (lift-resp resp xs≈ys pxs) -- List membership. infix 4 _∈_ _∉_ _∈_ : A → List A → Set x ∈ xs = Any (_≈_ x) xs _∉_ : A → List A → Set x ∉ xs = ¬ x ∈ xs -- Subsets. infix 4 _⊆_ _⊈_ _⊆_ : List A → List A → Set xs ⊆ ys = ∀ {x} → x ∈ xs → x ∈ ys _⊈_ : List A → List A → Set xs ⊈ ys = ¬ xs ⊆ ys -- Equality is respected by the predicate which is used to define -- _∈_. ∈-resp-≈ : ∀ {x} → (_≈_ x) Respects _≈_ ∈-resp-≈ = flip S.trans -- List equality is respected by _∈_. ∈-resp-list-≈ : ∀ {x} → _∈_ x Respects _≋_ ∈-resp-list-≈ = lift-resp ∈-resp-≈ -- _⊆_ is a preorder. ⊆-preorder : Preorder ⊆-preorder = record { carrier = List A ; _≈_ = _≋_ ; _∼_ = _⊆_ ; isPreorder = record { isEquivalence = Setoid.isEquivalence L.setoid ; reflexive = reflexive ; trans = λ ys⊆zs xs⊆ys → xs⊆ys ∘ ys⊆zs ; ∼-resp-≈ = (λ ys₁≋ys₂ xs⊆ys₁ → reflexive ys₁≋ys₂ ∘ xs⊆ys₁) , (λ xs₁≋xs₂ xs₁⊆ys → xs₁⊆ys ∘ reflexive (L.sym xs₁≋xs₂)) } } where reflexive : _≋_ ⇒ _⊆_ reflexive eq = ∈-resp-list-≈ eq module ⊆-Reasoning where import Relation.Binary.PreorderReasoning as PreR open PreR ⊆-preorder public renaming (_∼⟨_⟩_ to _⊆⟨_⟩_) infix 1 _∈⟨_⟩_ _∈⟨_⟩_ : ∀ x {xs ys} → x ∈ xs → xs IsRelatedTo ys → x ∈ ys x ∈⟨ x∈xs ⟩ xs⊆ys = (begin xs⊆ys) x∈xs -- A variant of List.map. map-with-∈ : ∀ {B} (xs : List A) → (∀ {x} → x ∈ xs → B) → List B map-with-∈ [] f = [] map-with-∈ (x ∷ xs) f = f (here S.refl) ∷ map-with-∈ xs (f ∘ there) -- Finds an element satisfying the predicate. find : ∀ {P : A → Set} {xs} → Any P xs → ∃ λ x → x ∈ xs × P x find (here px) = (_ , here S.refl , px) find (there pxs) = Prod.map id (Prod.map there id) (find pxs) lose : ∀ {P x xs} → P Respects _≈_ → x ∈ xs → P x → Any P xs lose resp x∈xs px = map (flip resp px) x∈xs -- The code above instantiated (and slightly changed) for -- propositional equality. module Membership-≡ {A : Set} where private open module M = Membership (PropEq.setoid A) public hiding (lift-resp; lose; ⊆-preorder; module ⊆-Reasoning) lose : ∀ {P x xs} → x ∈ xs → P x → Any P xs lose {P} = M.lose (PropEq.subst P) -- _⊆_ is a preorder. ⊆-preorder : Preorder ⊆-preorder = record { carrier = List A ; _≈_ = _≡_ ; _∼_ = _⊆_ ; isPreorder = record { isEquivalence = PropEq.isEquivalence ; reflexive = λ eq → PropEq.subst (_∈_ _) eq ; trans = λ ys⊆zs xs⊆ys → xs⊆ys ∘ ys⊆zs ; ∼-resp-≈ = PropEq.resp₂ _⊆_ } } module ⊆-Reasoning where import Relation.Binary.PreorderReasoning as PreR open PreR ⊆-preorder public renaming (_∼⟨_⟩_ to _⊆⟨_⟩_; _≈⟨_⟩_ to _≡⟨_⟩_) infix 1 _∈⟨_⟩_ _∈⟨_⟩_ : ∀ x {xs ys} → x ∈ xs → xs IsRelatedTo ys → x ∈ ys x ∈⟨ x∈xs ⟩ xs⊆ys = (begin xs⊆ys) x∈xs ------------------------------------------------------------------------ -- Another function -- If any element satisfies P, then P is satisfied. satisfied : ∀ {A} {P : A → Set} {xs} → Any P xs → ∃ P satisfied = Prod.map id Prod.proj₂ ∘ Membership-≡.find