------------------------------------------------------------------------ -- The Agda standard library -- -- Propositional equality -- -- This file contains some core properies of propositional equality -- which are re-exported by Relation.Binary.PropositionalEquality. They -- are ``equality rearrangement'' lemmas. ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --safe #-} module Relation.Binary.PropositionalEquality.Properties where open import Function.Base using (id; _∘_) open import Level using (Level) open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder; Poset) open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder) open import Relation.Binary.Definitions using (DecidableEquality) import Relation.Binary.Properties.Setoid as Setoid using (≈-isPreorder; ≈-isPartialOrder; ≈-preorder; ≈-poset) open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; refl; subst; trans; sym; subst₂; cong₂) open import Relation.Binary.Reasoning.Syntax open import Relation.Unary using (Pred) private variable a b c p : Level A B C : Set a ------------------------------------------------------------------------ -- Standard eliminator for the propositional equality type J : {A : Set a} {x : A} (B : (y : A) → x ≡ y → Set b) {y : A} (p : x ≡ y) → B x refl → B y p J B refl b = b ------------------------------------------------------------------------ -- Binary and/or dependent versions of standard operations on equality dcong : ∀ {A : Set a} {B : A → Set b} (f : (x : A) → B x) {x y} → (p : x ≡ y) → subst B p (f x) ≡ f y dcong f refl = refl dcong₂ : ∀ {A : Set a} {B : A → Set b} {C : Set c} (f : (x : A) → B x → C) {x₁ x₂ y₁ y₂} → (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂ → f x₁ y₁ ≡ f x₂ y₂ dcong₂ f refl refl = refl dsubst₂ : ∀ {A : Set a} {B : A → Set b} (C : (x : A) → B x → Set c) {x₁ x₂ y₁ y₂} (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂ → C x₁ y₁ → C x₂ y₂ dsubst₂ C refl refl c = c ddcong₂ : ∀ {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} (f : (x : A) (y : B x) → C x y) {x₁ x₂ y₁ y₂} (p : x₁ ≡ x₂) (q : subst B p y₁ ≡ y₂) → dsubst₂ C p q (f x₁ y₁) ≡ f x₂ y₂ ddcong₂ f refl refl = refl ------------------------------------------------------------------------ -- Various equality rearrangement lemmas trans-reflʳ : ∀ {x y : A} (p : x ≡ y) → trans p refl ≡ p trans-reflʳ refl = refl trans-assoc : ∀ {x y z u : A} (p : x ≡ y) {q : y ≡ z} {r : z ≡ u} → trans (trans p q) r ≡ trans p (trans q r) trans-assoc refl = refl trans-symˡ : ∀ {x y : A} (p : x ≡ y) → trans (sym p) p ≡ refl trans-symˡ refl = refl trans-symʳ : ∀ {x y : A} (p : x ≡ y) → trans p (sym p) ≡ refl trans-symʳ refl = refl trans-injectiveˡ : ∀ {x y z : A} {p₁ p₂ : x ≡ y} (q : y ≡ z) → trans p₁ q ≡ trans p₂ q → p₁ ≡ p₂ trans-injectiveˡ refl = subst₂ _≡_ (trans-reflʳ _) (trans-reflʳ _) trans-injectiveʳ : ∀ {x y z : A} (p : x ≡ y) {q₁ q₂ : y ≡ z} → trans p q₁ ≡ trans p q₂ → q₁ ≡ q₂ trans-injectiveʳ refl eq = eq cong-id : ∀ {x y : A} (p : x ≡ y) → cong id p ≡ p cong-id refl = refl cong-∘ : ∀ {x y : A} {f : B → C} {g : A → B} (p : x ≡ y) → cong (f ∘ g) p ≡ cong f (cong g p) cong-∘ refl = refl sym-cong : ∀ {x y : A} {f : A → B} (p : x ≡ y) → sym (cong f p) ≡ cong f (sym p) sym-cong refl = refl trans-cong : ∀ {x y z : A} {f : A → B} (p : x ≡ y) {q : y ≡ z} → trans (cong f p) (cong f q) ≡ cong f (trans p q) trans-cong refl = refl cong₂-reflˡ : ∀ {_∙_ : A → B → C} {x u v} → (p : u ≡ v) → cong₂ _∙_ refl p ≡ cong (x ∙_) p cong₂-reflˡ refl = refl cong₂-reflʳ : ∀ {_∙_ : A → B → C} {x y u} → (p : x ≡ y) → cong₂ _∙_ p refl ≡ cong (_∙ u) p cong₂-reflʳ refl = refl module _ {P : Pred A p} {x y : A} where subst-injective : ∀ (x≡y : x ≡ y) {p q : P x} → subst P x≡y p ≡ subst P x≡y q → p ≡ q subst-injective refl p≡q = p≡q subst-subst : ∀ {z} (x≡y : x ≡ y) {y≡z : y ≡ z} {p : P x} → subst P y≡z (subst P x≡y p) ≡ subst P (trans x≡y y≡z) p subst-subst refl = refl subst-subst-sym : (x≡y : x ≡ y) {p : P y} → subst P x≡y (subst P (sym x≡y) p) ≡ p subst-subst-sym refl = refl subst-sym-subst : (x≡y : x ≡ y) {p : P x} → subst P (sym x≡y) (subst P x≡y p) ≡ p subst-sym-subst refl = refl subst-∘ : ∀ {x y : A} {P : Pred B p} {f : A → B} (x≡y : x ≡ y) {p : P (f x)} → subst (P ∘ f) x≡y p ≡ subst P (cong f x≡y) p subst-∘ refl = refl -- Lemma 2.3.11 in the HoTT book, and `transport_map` in the UniMath -- library subst-application′ : ∀ {a b₁ b₂} {A : Set a} (B₁ : A → Set b₁) {B₂ : A → Set b₂} {x₁ x₂ : A} {y : B₁ x₁} (g : ∀ x → B₁ x → B₂ x) (eq : x₁ ≡ x₂) → subst B₂ eq (g x₁ y) ≡ g x₂ (subst B₁ eq y) subst-application′ _ _ refl = refl subst-application : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂} (B₁ : A₁ → Set b₁) {B₂ : A₂ → Set b₂} {f : A₂ → A₁} {x₁ x₂ : A₂} {y : B₁ (f x₁)} (g : ∀ x → B₁ (f x) → B₂ x) (eq : x₁ ≡ x₂) → subst B₂ eq (g x₁ y) ≡ g x₂ (subst B₁ (cong f eq) y) subst-application _ _ refl = refl ------------------------------------------------------------------------ -- Structure of equality as a binary relation isEquivalence : IsEquivalence {A = A} _≡_ isEquivalence = record { refl = refl ; sym = sym ; trans = trans } isDecEquivalence : DecidableEquality A → IsDecEquivalence _≡_ isDecEquivalence _≟_ = record { isEquivalence = isEquivalence ; _≟_ = _≟_ } setoid : Set a → Setoid _ _ setoid A = record { Carrier = A ; _≈_ = _≡_ ; isEquivalence = isEquivalence } decSetoid : DecidableEquality A → DecSetoid _ _ decSetoid _≟_ = record { _≈_ = _≡_ ; isDecEquivalence = isDecEquivalence _≟_ } ------------------------------------------------------------------------ -- Bundles for equality as a binary relation isPreorder : IsPreorder {A = A} _≡_ _≡_ isPreorder = Setoid.≈-isPreorder (setoid _) isPartialOrder : IsPartialOrder {A = A} _≡_ _≡_ isPartialOrder = Setoid.≈-isPartialOrder (setoid _) preorder : Set a → Preorder _ _ _ preorder A = Setoid.≈-preorder (setoid A) poset : Set a → Poset _ _ _ poset A = Setoid.≈-poset (setoid A) ------------------------------------------------------------------------ -- Reasoning -- This is a special instance of `Relation.Binary.Reasoning.Setoid`. -- Rather than instantiating the latter with (setoid A), we reimplement -- equation chains from scratch since then goals are printed much more -- readably. module ≡-Reasoning {a} {A : Set a} where open begin-syntax {A = A} _≡_ id public open ≡-syntax {A = A} _≡_ trans public open end-syntax {A = A} _≡_ refl public