-- Martin Escardo, 3rd August 2015. {-# OPTIONS --without-K #-} module prop where open import preliminaries open import isprop public -- The type of small propositions is large: Prp : {i : 𝕃} → U(lsuc i) Prp{i} = Σ \(P : U i) → isProp P -- Destructors: _holds : {i : 𝕃} → Prp{i} → U i _holds = pr₁ holdsIsProp : {i : 𝕃} (p : Prp{i}) → isProp(p holds) holdsIsProp = pr₂ -- The idea of the terminology is that we cannot assert a point of the -- type Prop, as it is a pair, but we can assert that it holds, meaning -- that we assert that its first component has an inhabitant, as it is -- a truth-value (a type P with the asserted side-condition isProp P). -- We have the following judgemental equalities: -- -- (β₁) ( P , isp ) holds = P, -- (β₂) holdsIsProp ( P , isp ) = isp, -- (η) p = ( p holds , holdsIsProp p ). -- NB. holdsIsProp shows that _holds is an embedding in the sense of -- the HoTT book https://homotopytypetheory.org/book/. Prop-≡ : {i : 𝕃} {p q : Prp{i}} → p holds ≡ q holds → p ≡ q Prop-≡ {i} {p} {q} e = e' where e' : p ≡ q e' = Σ-≡ e (isProp-isProp (transport isProp e (holdsIsProp p)) (holdsIsProp q)) -- Propositional univalence says that any two (logically) equivalent -- propositions are equal: postulate propua : {i : 𝕃} {P Q : U i} → isProp P → isProp Q → (P → Q) → (Q → P) → P ≡ Q -- It can be formulated as follows: propext : {i : 𝕃} {p q : Prp{i}} → (p holds → q holds) → (q holds → p holds) → p ≡ q propext {i} {p} {q} f g = Prop-≡ (propua (holdsIsProp p) (holdsIsProp q) f g)