------------------------------------------------------------------------ -- The Agda standard library -- -- Propositional (intensional) equality ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} module Relation.Binary.PropositionalEquality where import Axiom.Extensionality.Propositional as Ext open import Axiom.UniquenessOfIdentityProofs open import Function.Base using (id; _∘_) open import Function.Equality using (Π; _⟶_; ≡-setoid) open import Level using (Level; _⊔_) open import Data.Product using (∃) open import Relation.Nullary using (yes ; no) open import Relation.Nullary.Decidable.Core open import Relation.Binary open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid) import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial as Trivial private variable a b c ℓ p : Level A : Set a B : Set b C : Set c ------------------------------------------------------------------------ -- Re-export contents modules that make up the parts open import Relation.Binary.PropositionalEquality.Core public open import Relation.Binary.PropositionalEquality.Properties public open import Relation.Binary.PropositionalEquality.Algebra public ------------------------------------------------------------------------ -- Pointwise equality infix 4 _≗_ _→-setoid_ : ∀ (A : Set a) (B : Set b) → Setoid _ _ A →-setoid B = ≡-setoid A (Trivial.indexedSetoid (setoid B)) _≗_ : (f g : A → B) → Set _ _≗_ {A = A} {B = B} = Setoid._≈_ (A →-setoid B) :→-to-Π : ∀ {A : Set a} {B : IndexedSetoid A b ℓ} → ((x : A) → IndexedSetoid.Carrier B x) → Π (setoid A) B :→-to-Π {B = B} f = record { _⟨$⟩_ = f ; cong = λ { refl → IndexedSetoid.refl B } } where open IndexedSetoid B using (_≈_) →-to-⟶ : ∀ {A : Set a} {B : Setoid b ℓ} → (A → Setoid.Carrier B) → setoid A ⟶ B →-to-⟶ = :→-to-Π ------------------------------------------------------------------------ -- Inspect -- Inspect can be used when you want to pattern match on the result r -- of some expression e, and you also need to "remember" that r ≡ e. -- See README.Inspect for an explanation of how/why to use this. record Reveal_·_is_ {A : Set a} {B : A → Set b} (f : (x : A) → B x) (x : A) (y : B x) : Set (a ⊔ b) where constructor [_] field eq : f x ≡ y inspect : ∀ {A : Set a} {B : A → Set b} (f : (x : A) → B x) (x : A) → Reveal f · x is f x inspect f x = [ refl ] ------------------------------------------------------------------------ -- Propositionality isPropositional : Set a → Set a isPropositional A = (a b : A) → a ≡ b ------------------------------------------------------------------------ -- More complex rearrangement lemmas -- A lemma that is very similar to Lemma 2.4.3 from the HoTT book. naturality : ∀ {x y} {x≡y : x ≡ y} {f g : A → B} (f≡g : ∀ x → f x ≡ g x) → trans (cong f x≡y) (f≡g y) ≡ trans (f≡g x) (cong g x≡y) naturality {x = x} {x≡y = refl} f≡g = f≡g x ≡⟨ sym (trans-reflʳ _) ⟩ trans (f≡g x) refl ∎ where open ≡-Reasoning -- A lemma that is very similar to Corollary 2.4.4 from the HoTT book. cong-≡id : ∀ {f : A → A} {x : A} (f≡id : ∀ x → f x ≡ x) → cong f (f≡id x) ≡ f≡id (f x) cong-≡id {f = f} {x} f≡id = cong f fx≡x ≡⟨ sym (trans-reflʳ _) ⟩ trans (cong f fx≡x) refl ≡⟨ cong (trans _) (sym (trans-symʳ fx≡x)) ⟩ trans (cong f fx≡x) (trans fx≡x (sym fx≡x)) ≡⟨ sym (trans-assoc (cong f fx≡x)) ⟩ trans (trans (cong f fx≡x) fx≡x) (sym fx≡x) ≡⟨ cong (λ p → trans p (sym _)) (naturality f≡id) ⟩ trans (trans f²x≡x (cong id fx≡x)) (sym fx≡x) ≡⟨ cong (λ p → trans (trans f²x≡x p) (sym fx≡x)) (cong-id _) ⟩ trans (trans f²x≡x fx≡x) (sym fx≡x) ≡⟨ trans-assoc f²x≡x ⟩ trans f²x≡x (trans fx≡x (sym fx≡x)) ≡⟨ cong (trans _) (trans-symʳ fx≡x) ⟩ trans f²x≡x refl ≡⟨ trans-reflʳ _ ⟩ f≡id (f x) ∎ where open ≡-Reasoning; fx≡x = f≡id x; f²x≡x = f≡id (f x) module _ (_≟_ : Decidable {A = A} _≡_) {x y : A} where ≡-≟-identity : (eq : x ≡ y) → x ≟ y ≡ yes eq ≡-≟-identity eq = dec-yes-irr (x ≟ y) (Decidable⇒UIP.≡-irrelevant _≟_) eq ≢-≟-identity : x ≢ y → ∃ λ ¬eq → x ≟ y ≡ no ¬eq ≢-≟-identity ¬eq = dec-no (x ≟ y) ¬eq ------------------------------------------------------------------------ -- DEPRECATED NAMES ------------------------------------------------------------------------ -- Please use the new names as continuing support for the old names is -- not guaranteed. -- Version 1.0 Extensionality = Ext.Extensionality {-# WARNING_ON_USAGE Extensionality "Warning: Extensionality was deprecated in v1.0. Please use Extensionality from `Axiom.Extensionality.Propositional` instead." #-} extensionality-for-lower-levels = Ext.lower-extensionality {-# WARNING_ON_USAGE extensionality-for-lower-levels "Warning: extensionality-for-lower-levels was deprecated in v1.0. Please use lower-extensionality from `Axiom.Extensionality.Propositional` instead." #-} ∀-extensionality = Ext.∀-extensionality {-# WARNING_ON_USAGE ∀-extensionality "Warning: ∀-extensionality was deprecated in v1.0. Please use ∀-extensionality from `Axiom.Extensionality.Propositional` instead." #-}