------------------------------------------------------------------------ -- The Agda standard library -- -- Properties of products ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --safe #-} module Data.Product.Properties where open import Axiom.UniquenessOfIdentityProofs using (UIP; module Decidable⇒UIP) open import Data.Product.Base using (_,_; Σ; _×_; map; swap; ∃; ∃₂) open import Function.Base using (_∋_; _∘_; id) open import Function.Bundles using (_↔_; mk↔ₛ′) open import Level using (Level) open import Relation.Binary.Definitions using (DecidableEquality) open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; _≗_; subst; cong; cong₂; cong′) open import Relation.Nullary.Decidable as Dec using (Dec; yes; no) private variable a b c d ℓ : Level A : Set a B : Set b C : Set c D : Set d ------------------------------------------------------------------------ -- Equality (dependent) module _ {B : A → Set b} where ,-injectiveˡ : ∀ {a c} {b : B a} {d : B c} → (a , b) ≡ (c , d) → a ≡ c ,-injectiveˡ refl = refl ,-injectiveʳ-≡ : ∀ {a b} {c : B a} {d : B b} → UIP A → (a , c) ≡ (b , d) → (q : a ≡ b) → subst B q c ≡ d ,-injectiveʳ-≡ {c = c} u refl q = cong (λ x → subst B x c) (u q refl) ,-injectiveʳ-UIP : ∀ {a} {b c : B a} → UIP A → (Σ A B ∋ (a , b)) ≡ (a , c) → b ≡ c ,-injectiveʳ-UIP u p = ,-injectiveʳ-≡ u p refl ≡-dec : DecidableEquality A → (∀ {a} → DecidableEquality (B a)) → DecidableEquality (Σ A B) ≡-dec dec₁ dec₂ (a , x) (b , y) with dec₁ a b ... | no [a≢b] = no ([a≢b] ∘ ,-injectiveˡ) ... | yes refl = Dec.map′ (cong (a ,_)) (,-injectiveʳ-UIP (Decidable⇒UIP.≡-irrelevant dec₁)) (dec₂ x y) ------------------------------------------------------------------------ -- Equality (non-dependent) ,-injectiveʳ : ∀ {a c : A} {b d : B} → (a , b) ≡ (c , d) → b ≡ d ,-injectiveʳ refl = refl ,-injective : ∀ {a c : A} {b d : B} → (a , b) ≡ (c , d) → a ≡ c × b ≡ d ,-injective refl = refl , refl map-cong : ∀ {f g : A → C} {h i : B → D} → f ≗ g → h ≗ i → map f h ≗ map g i map-cong f≗g h≗i (x , y) = cong₂ _,_ (f≗g x) (h≗i y) -- The following properties are definitionally true (because of η) -- but for symmetry with ⊎ it is convenient to define and name them. swap-involutive : swap {A = A} {B = B} ∘ swap ≗ id swap-involutive _ = refl ------------------------------------------------------------------------ -- Equality between pairs can be expressed as a pair of equalities module _ {A : Set a} {B : A → Set b} {p₁@(a₁ , b₁) p₂@(a₂ , b₂) : Σ A B} where Σ-≡,≡→≡ : Σ (a₁ ≡ a₂) (λ p → subst B p b₁ ≡ b₂) → p₁ ≡ p₂ Σ-≡,≡→≡ (refl , refl) = refl Σ-≡,≡←≡ : p₁ ≡ p₂ → Σ (a₁ ≡ a₂) (λ p → subst B p b₁ ≡ b₂) Σ-≡,≡←≡ refl = refl , refl private left-inverse-of : (p : Σ (a₁ ≡ a₂) (λ x → subst B x b₁ ≡ b₂)) → Σ-≡,≡←≡ (Σ-≡,≡→≡ p) ≡ p left-inverse-of (refl , refl) = refl right-inverse-of : (p : p₁ ≡ p₂) → Σ-≡,≡→≡ (Σ-≡,≡←≡ p) ≡ p right-inverse-of refl = refl Σ-≡,≡↔≡ : (∃ λ (p : a₁ ≡ a₂) → subst B p b₁ ≡ b₂) ↔ p₁ ≡ p₂ Σ-≡,≡↔≡ = mk↔ₛ′ Σ-≡,≡→≡ Σ-≡,≡←≡ right-inverse-of left-inverse-of -- the non-dependent case. Proofs are exactly as above, and straightforward. module _ {p₁@(a₁ , b₁) p₂@(a₂ , b₂) : A × B} where ×-≡,≡→≡ : (a₁ ≡ a₂ × b₁ ≡ b₂) → p₁ ≡ p₂ ×-≡,≡→≡ (refl , refl) = refl ×-≡,≡←≡ : p₁ ≡ p₂ → (a₁ ≡ a₂ × b₁ ≡ b₂) ×-≡,≡←≡ refl = refl , refl ×-≡,≡↔≡ : (a₁ ≡ a₂ × b₁ ≡ b₂) ↔ p₁ ≡ p₂ ×-≡,≡↔≡ = mk↔ₛ′ ×-≡,≡→≡ ×-≡,≡←≡ (λ { refl → refl }) (λ { (refl , refl) → refl }) ------------------------------------------------------------------------ -- The order of ∃₂ can be swapped ∃∃↔∃∃ : (R : A → B → Set ℓ) → (∃₂ λ x y → R x y) ↔ (∃₂ λ y x → R x y) ∃∃↔∃∃ R = mk↔ₛ′ to from cong′ cong′ where to : (∃₂ λ x y → R x y) → (∃₂ λ y x → R x y) to (x , y , Rxy) = (y , x , Rxy) from : (∃₂ λ y x → R x y) → (∃₂ λ x y → R x y) from (y , x , Rxy) = (x , y , Rxy)