{- Based on Nicolai Kraus' blog post: The Truncation Map |_| : ℕ -> ‖ℕ‖ is nearly Invertible https://homotopytypetheory.org/2013/10/28/the-truncation-map-_-ℕ-‖ℕ‖-is-nearly-invertible/ Defines [recover], which definitionally satisfies `recover ∣ x ∣ ≡ x` ([recover∣∣]) for homogeneous types Also see the follow-up post by Jason Gross: Composition is not what you think it is! Why “nearly invertible” isn’t. https://homotopytypetheory.org/2014/02/24/composition-is-not-what-you-think-it-is-why-nearly-invertible-isnt/ -} {-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.HITs.PropositionalTruncation.MagicTrick where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Function open import Cubical.Foundations.Path open import Cubical.Foundations.Pointed open import Cubical.Foundations.Pointed.Homogeneous open import Cubical.HITs.PropositionalTruncation.Base open import Cubical.HITs.PropositionalTruncation.Properties module Recover {ℓ} (A∙ : Pointed ℓ) (h : isHomogeneous A∙) where private A = typ A∙ a = pt A∙ toEquivPtd : ∥ A ∥ → Σ[ B∙ ∈ Pointed ℓ ] (A , a) ≡ B∙ toEquivPtd = rec (isContr→isProp (isContrSingl _)) (λ x → (A , x) , h x) private B∙ : ∥ A ∥ → Pointed ℓ B∙ tx = fst (toEquivPtd tx) -- the key observation is that B∙ ∣ x ∣ is definitionally equal to (A , x) private obvs : ∀ x → B∙ ∣ x ∣ ≡ (A , x) obvs x = refl -- try it: `C-c C-n B∙ ∣ x ∣` gives `(A , x)` -- thus any truncated element (of a homogeneous type) can be recovered by agda's normalizer! recover : ∀ (tx : ∥ A ∥) → typ (B∙ tx) recover tx = pt (B∙ tx) recover∣∣ : ∀ (x : A) → recover ∣ x ∣ ≡ x recover∣∣ x = refl -- try it: `C-c C-n recover ∣ x ∣` gives `x` private -- notice that the following typechecks because typ (B∙ ∣ x ∣) is definitionally equal to to A, but -- `recover : ∥ A ∥ → A` does not because typ (B∙ tx) is not definitionally equal to A (though it is -- judegmentally equal to A by cong typ (snd (toEquivPtd tx)) : A ≡ typ (B∙ tx)) obvs2 : A → A obvs2 = recover ∘ ∣_∣ -- one might wonder if (cong recover (squash ∣ x ∣ ∣ y ∣)) therefore has type x ≡ y, but thankfully -- typ (B∙ (squash ∣ x ∣ ∣ y ∣ i)) is *not* A (it's a messy hcomp involving h x and h y) recover-squash : ∀ x y → -- x ≡ y -- this raises an error PathP (λ i → typ (B∙ (squash ∣ x ∣ ∣ y ∣ i))) x y recover-squash x y = cong recover (squash ∣ x ∣ ∣ y ∣) -- Demo, adapted from: -- https://bitbucket.org/nicolaikraus/agda/src/e30d70c72c6af8e62b72eefabcc57623dd921f04/trunc-inverse.lagda private open import Cubical.Data.Nat open Recover (ℕ , zero) (isHomogeneousDiscrete discreteℕ) -- only `∣hidden∣` is exported, `hidden` is no longer in scope module _ where private hidden : ℕ hidden = 17 ∣hidden∣ : ∥ ℕ ∥ ∣hidden∣ = ∣ hidden ∣ -- we can still recover the value, even though agda can no longer see `hidden`! test : recover ∣hidden∣ ≡ 17 test = refl -- try it: `C-c C-n recover ∣hidden∣` gives `17` -- `C-c C-n hidden` gives an error -- Finally, note that the definition of recover is independent of the proof that A is homogeneous. Thus we -- still can definitionally recover information hidden by ∣_∣ as long as we permit holes. Try replacing -- `isHomogeneousDiscrete discreteℕ` above with a hole (`?`) and notice that everything still works

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