⚠️ This module is not part of this project and is only included for reference.
It is either part of the 1lab, the cubical library, or a built-in Agda module.
--- definition: | We construct propositional truncations, the reflections of a type into the universe of propositions. --- <!-- ```agda open import 1Lab.Reflection.Induction open import 1Lab.Reflection.HLevel open import 1Lab.HLevel.Retracts open import 1Lab.Path.Reasoning open import 1Lab.Type.Sigma open import 1Lab.HLevel open import 1Lab.Equiv open import 1Lab.Path open import 1Lab.Type ``` --> ```agda module 1Lab.HIT.Truncation where ``` # Propositional truncation {defines="propositional-truncation"} Let $A$ be a type. The **propositional truncation** of $A$ is a type which represents the [[proposition]] "A is inhabited". In MLTT, propositional truncations can not be constructed without postulates, even in the presence of impredicative prop. However, Cubical Agda provides a tool to define them: _higher inductive types_. ```agda data ∥_∥ {ℓ} (A : Type ℓ) : Type ℓ where inc : A → ∥ A ∥ squash : is-prop ∥ A ∥ ``` The two constructors that generate `∥_∥`{.Agda} state precisely that the truncation is inhabited when `A` is (`inc`{.Agda}), and that it is a proposition (`squash`{.Agda}). ```agda is-prop-∥-∥ : ∀ {ℓ} {A : Type ℓ} → is-prop ∥ A ∥ is-prop-∥-∥ = squash ``` <!-- ```agda instance H-Level-truncation : ∀ {n} {ℓ} {A : Type ℓ} → H-Level ∥ A ∥ (suc n) H-Level-truncation = prop-instance squash ``` --> The eliminator for `∥_∥`{.Agda} says that you can eliminate onto $P$ whenever it is a family of propositions, by providing a case for `inc`{.Agda}. ```agda ∥-∥-elim : ∀ {ℓ ℓ'} {A : Type ℓ} {P : ∥ A ∥ → Type ℓ'} → ((x : _) → is-prop (P x)) → ((x : A) → P (inc x)) → (x : ∥ A ∥) → P x ∥-∥-elim pprop incc (inc x) = incc x ∥-∥-elim pprop incc (squash x y i) = is-prop→pathp (λ j → pprop (squash x y j)) (∥-∥-elim pprop incc x) (∥-∥-elim pprop incc y) i ``` <!-- ```agda ∥-∥-elim₂ : ∀ {ℓ ℓ₁ ℓ₂} {A : Type ℓ} {B : Type ℓ₁} {P : ∥ A ∥ → ∥ B ∥ → Type ℓ₂} → (∀ x y → is-prop (P x y)) → (∀ x y → P (inc x) (inc y)) → ∀ x y → P x y ∥-∥-elim₂ {A = A} {B} {P} pprop work x y = go x y where go : ∀ x y → P x y go (inc x) (inc x₁) = work x x₁ go (inc x) (squash y y₁ i) = is-prop→pathp (λ i → pprop (inc x) (squash y y₁ i)) (go (inc x) y) (go (inc x) y₁) i go (squash x x₁ i) z = is-prop→pathp (λ i → pprop (squash x x₁ i) z) (go x z) (go x₁ z) i ∥-∥-rec : ∀ {ℓ ℓ'} {A : Type ℓ} {P : Type ℓ'} → is-prop P → (A → P) → (x : ∥ A ∥) → P ∥-∥-rec pprop = ∥-∥-elim (λ _ → pprop) ∥-∥-proj : ∀ {ℓ} {A : Type ℓ} → is-prop A → ∥ A ∥ → A ∥-∥-proj ap = ∥-∥-rec ap λ x → x ∥-∥-rec₂ : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ''} {P : Type ℓ'} → is-prop P → (A → B → P) → (x : ∥ A ∥) (y : ∥ B ∥) → P ∥-∥-rec₂ pprop = ∥-∥-elim₂ (λ _ _ → pprop) ∥-∥-rec! : ∀ {ℓ ℓ'} {A : Type ℓ} {P : Type ℓ'} → {@(tactic hlevel-tactic-worker) pprop : is-prop P} → (A → P) → (x : ∥ A ∥) → P ∥-∥-rec! {pprop = pprop} = ∥-∥-elim (λ _ → pprop) ∥-∥-proj! : ∀ {ℓ} {A : Type ℓ} → {@(tactic hlevel-tactic-worker) ap : is-prop A} → ∥ A ∥ → A ∥-∥-proj! {ap = ap} = ∥-∥-proj ap ``` --> The propositional truncation can be called the **free proposition** on a type, because it satisfies the universal property that a [[left adjoint]] would have. Specifically, let `B` be a proposition. We have: ```agda ∥-∥-univ : ∀ {ℓ} {A : Type ℓ} {B : Type ℓ} → is-prop B → (∥ A ∥ → B) ≃ (A → B) ∥-∥-univ {A = A} {B = B} bprop = Iso→Equiv (inc' , iso rec (λ _ → refl) beta) where inc' : (x : ∥ A ∥ → B) → A → B inc' f x = f (inc x) rec : (f : A → B) → ∥ A ∥ → B rec f (inc x) = f x rec f (squash x y i) = bprop (rec f x) (rec f y) i beta : _ beta f = funext (∥-∥-elim (λ _ → is-prop→is-set bprop _ _) (λ _ → refl)) ``` Furthermore, as required of a free construction, the propositional truncation extends to a functor: ```agda ∥-∥-map : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → (A → B) → ∥ A ∥ → ∥ B ∥ ∥-∥-map f (inc x) = inc (f x) ∥-∥-map f (squash x y i) = squash (∥-∥-map f x) (∥-∥-map f y) i ``` <!-- ```agda ∥-∥-map₂ : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} → (A → B → C) → ∥ A ∥ → ∥ B ∥ → ∥ C ∥ ∥-∥-map₂ f (inc x) (inc y) = inc (f x y) ∥-∥-map₂ f (squash x y i) z = squash (∥-∥-map₂ f x z) (∥-∥-map₂ f y z) i ∥-∥-map₂ f x (squash y z i) = squash (∥-∥-map₂ f x y) (∥-∥-map₂ f x z) i ``` --> Using the propositional truncation, we can define the **existential quantifier** as a truncated `Σ`{.Agda}. ```agda ∃ : ∀ {a b} (A : Type a) (B : A → Type b) → Type _ ∃ A B = ∥ Σ A B ∥ syntax ∃ A (λ x → B) = ∃[ x ∈ A ] B ``` Note that if $P$ is already a proposition, then truncating it does nothing: ```agda is-prop→equiv∥-∥ : ∀ {ℓ} {P : Type ℓ} → is-prop P → P ≃ ∥ P ∥ is-prop→equiv∥-∥ pprop = prop-ext pprop squash inc (∥-∥-proj pprop) ``` In fact, an alternative definition of `is-prop`{.Agda} is given by "being equivalent to your own truncation": ```agda is-prop≃equiv∥-∥ : ∀ {ℓ} {P : Type ℓ} → is-prop P ≃ (P ≃ ∥ P ∥) is-prop≃equiv∥-∥ {P = P} = prop-ext is-prop-is-prop eqv-prop is-prop→equiv∥-∥ inv where inv : (P ≃ ∥ P ∥) → is-prop P inv eqv = equiv→is-hlevel 1 ((eqv e⁻¹) .fst) ((eqv e⁻¹) .snd) squash eqv-prop : is-prop (P ≃ ∥ P ∥) eqv-prop x y = Σ-path (λ i p → squash (x .fst p) (y .fst p) i) (is-equiv-is-prop _ _ _) ``` :::{.definition #merely alias="mere"} Throughout the 1Lab, we use the words "mere" and "merely" to modify a type to mean its propositional truncation. This terminology is adopted from the HoTT book. For example, a type $X$ is said _merely equivalent_ to $Y$ if the type $\| X \equiv Y \|$ is inhabited. ::: ## Maps into sets The elimination principle for $\| A \|$ says that we can only use the $A$ inside in a way that _doesn't matter_: the motive of elimination must be a family of propositions, so our use of $A$ must not matter in a very strong sense. Often, it's useful to relax this requirement slightly: Can we map out of $\| A \|$ using a _constant_ function? The answer is yes, provided we're mapping into a [[set]]! In that case, the [[image]] of $f$ is a proposition, so that we can map from $\| A \| \to \im f \to B$. In the [next section](#maps-into-groupoids), we'll see a more general method for mapping into types that aren't sets. From the discussion in [1Lab.Counterexamples.Sigma], we know the definition of image, or more properly of $(-1)$-image: [1Lab.Counterexamples.Sigma]: 1Lab.Counterexamples.Sigma.html ```agda image : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → (A → B) → Type _ image {A = A} {B = B} f = Σ[ b ∈ B ] ∃[ a ∈ A ] (f a ≡ b) ``` To see that the `image`{.Agda} indeed implements the concept of image, we define a way to factor any map through its image. By the definition of image, we have that the map `f-image`{.Agda} is always surjective, and since `∃` is a family of props, the first projection out of `image`{.Agda} is an embedding. Thus we factor a map $f$ as $A \epi \im f \mono B$. ```agda f-image : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → (f : A → B) → A → image f f-image f x = f x , inc (x , refl) ``` We now prove the theorem that will let us map out of a propositional truncation using a constant function into sets: if $B$ is a set, and $f : A \to B$ is a constant function, then $\im f$ is a proposition. ```agda is-constant→image-is-prop : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → is-set B → (f : A → B) → (∀ x y → f x ≡ f y) → is-prop (image f) ``` This is intuitively true (if the function is constant, then there is at most one thing in the image!), but formalising it turns out to be slightly tricky, and the requirement that $B$ be a set is perhaps unexpected. A sketch of the proof is as follows. Suppose that we have some $(a, x)$ and $(b, y)$ in the image. We know, morally, that $x$ (respectively $y$) give us some $f^*(a) : A$ and $p : f(f^*a) = a$ (resp $q : f(f^*(b)) = b$) --- which would establish that $a \equiv b$, as we need, since we have $a = f(f^*(a)) = f(f^*(b)) = b$, where the middle equation is by constancy of $f$ --- but $p$ and $q$ are hidden under propositional truncations, so we crucially need to use the fact that $B$ is a set so that $a = b$ is a proposition. ```agda is-constant→image-is-prop bset f fconst (a , x) (b , y) = Σ-prop-path (λ _ → squash) (∥-∥-elim₂ (λ _ _ → bset _ _) (λ { (f*a , p) (f*b , q) → sym p ·· fconst f*a f*b ·· q }) x y) ``` Using the image factorisation, we can project from a propositional truncation onto a set using a constant map. ```agda ∥-∥-rec-set : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → is-set B → (f : A → B) → (∀ x y → f x ≡ f y) → ∥ A ∥ → B ∥-∥-rec-set {A = A} {B} bset f fconst x = ∥-∥-elim {P = λ _ → image f} (λ _ → is-constant→image-is-prop bset f fconst) (f-image f) x .fst ``` ## Maps into groupoids We can push this idea further: as discussed in [@Kraus:2015], in general, functions $\| A \| \to B$ are equivalent to **coherently constant** functions $A \to B$. This involves an infinite tower of conditions, each relating to the previous one, which isn't something we can easily formulate in the language of type theory. However, when $B$ is an $n$-type, it is enough to ask for the first $n$ levels of the tower. In the case of sets, we've [seen](#maps-into-sets) that the naïve notion of constancy is enough. We now deal with the case of [[groupoids]], which requires an additional step: we ask for a function $f : A \to B$ equipped with a witness of constancy $\rm{const}_{x,y} : f x \equiv f y$ *and* a coherence $\rm{coh}_{x,y,z} : \rm{const}_{x,y} \bullet \rm{const}_{y,z} \equiv \rm{const}_{x,z}$. This time, we cannot hope to show that the image of $f$ is a proposition: the image of a map $\top \to S^1$ is $S^1$. Instead, we use the following higher inductive type, which can be thought of as the "codiscrete groupoid" on $A$: ```agda data ∥_∥³ {ℓ} (A : Type ℓ) : Type ℓ where inc : A → ∥ A ∥³ iconst : ∀ a b → inc a ≡ inc b icoh : ∀ a b c → PathP (λ i → inc a ≡ iconst b c i) (iconst a b) (iconst a c) squash : is-groupoid ∥ A ∥³ ``` The recursion principle for this type says exactly that any coherently constant function into a groupoid factors through $\| A \|^3$! ```agda ∥-∥³-rec : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → is-groupoid B → (f : A → B) → (fconst : ∀ x y → f x ≡ f y) → (∀ x y z → fconst x y ∙ fconst y z ≡ fconst x z) → ∥ A ∥³ → B ∥-∥³-rec {A = A} {B} bgrpd f fconst fcoh = go where go : ∥ A ∥³ → B go (inc x) = f x go (iconst a b i) = fconst a b i go (icoh a b c i j) = ∙→square (sym (fcoh a b c)) i j go (squash x y a b p q i j k) = bgrpd (go x) (go y) (λ i → go (a i)) (λ i → go (b i)) (λ i j → go (p i j)) (λ i j → go (q i j)) i j k ``` All that remains to show is that $\| A \|^3$ is a proposition^[in fact, it's even a *propositional truncation* of $A$, in that it satisfies the same universal property as $\| A \|$], which mainly requires writing more elimination principles. <!-- ```agda ∥-∥³-elim-set : ∀ {ℓ} {A : Type ℓ} {ℓ'} {P : ∥ A ∥³ → Type ℓ'} → (∀ a → is-set (P a)) → (f : (a : A) → P (inc a)) → (∀ a b → PathP (λ i → P (iconst a b i)) (f a) (f b)) → ∀ a → P a unquoteDef ∥-∥³-elim-set = make-elim-n 2 ∥-∥³-elim-set (quote ∥_∥³) ∥-∥³-elim-prop : ∀ {ℓ} {A : Type ℓ} {ℓ'} {P : ∥ A ∥³ → Type ℓ'} → (∀ a → is-prop (P a)) → (f : (a : A) → P (inc a)) → ∀ a → P a unquoteDef ∥-∥³-elim-prop = make-elim-n 1 ∥-∥³-elim-prop (quote ∥_∥³) ``` --> ```agda ∥-∥³-is-prop : ∀ {ℓ} {A : Type ℓ} → is-prop ∥ A ∥³ ∥-∥³-is-prop = is-contr-if-inhabited→is-prop $ ∥-∥³-elim-prop (λ _ → hlevel 1) (λ a → contr (inc a) (∥-∥³-elim-set (λ _ → squash _ _) (iconst a) (icoh a))) ``` Hence we get the desired recursion principle for the usual propositional truncation. ```agda ∥-∥-rec-groupoid : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → is-groupoid B → (f : A → B) → (fconst : ∀ x y → f x ≡ f y) → (∀ x y z → fconst x y ∙ fconst y z ≡ fconst x z) → ∥ A ∥ → B ∥-∥-rec-groupoid bgrpd f fconst fcoh = ∥-∥³-rec bgrpd f fconst fcoh ∘ ∥-∥-rec ∥-∥³-is-prop inc ``` <details> <summary> As we hinted at above, this method generalises (externally) to $n$-types; we sketch the details of the next level for the curious reader. </summary> The next coherence involves a tetrahedron all of whose faces are $\rm{coh}$, or, since we're doing cubical type theory, a "cubical tetrahedron": ~~~{.quiver .tall-15} \[\begin{tikzcd} a &&& a \\ & b & b \\ & c & d \\ a &&& a \arrow[""{name=0, anchor=center, inner sep=0}, from=3-2, to=3-3] \arrow[""{name=1, anchor=center, inner sep=0}, Rightarrow, no head, from=2-2, to=2-3] \arrow[""{name=2, anchor=center, inner sep=0}, from=2-3, to=3-3] \arrow[from=2-2, to=3-2] \arrow[from=1-4, to=2-3] \arrow[""{name=3, anchor=center, inner sep=0}, from=1-1, to=2-2] \arrow[Rightarrow, no head, from=1-1, to=1-4] \arrow[""{name=4, anchor=center, inner sep=0}, from=4-4, to=3-3] \arrow[""{name=5, anchor=center, inner sep=0}, Rightarrow, no head, from=1-4, to=4-4] \arrow[""{name=6, anchor=center, inner sep=0}, from=4-1, to=3-2] \arrow[Rightarrow, no head, from=4-1, to=1-1] \arrow[Rightarrow, no head, from=4-1, to=4-4] \arrow["{\rm{coh}_{b,c,d}}"{description}, draw=none, from=0, to=1] \arrow["{\rm{coh}_{a,b,d}}"{description}, draw=none, from=2, to=5] \arrow["{\rm{coh}_{a,c,d}}"{description}, draw=none, from=6, to=4] \arrow["{\rm{coh}_{a,b,c}}"{description}, draw=none, from=3, to=6] \end{tikzcd}\] ~~~ ```agda data ∥_∥⁴ {ℓ} (A : Type ℓ) : Type ℓ where inc : A → ∥ A ∥⁴ iconst : ∀ a b → inc a ≡ inc b icoh : ∀ a b c → PathP (λ i → inc a ≡ iconst b c i) (iconst a b) (iconst a c) iassoc : ∀ a b c d → PathP (λ i → PathP (λ j → inc a ≡ icoh b c d i j) (iconst a b) (icoh a c d i)) (icoh a b c) (icoh a b d) squash : is-hlevel ∥ A ∥⁴ 4 ∥-∥⁴-rec : ∀ {ℓ} {A : Type ℓ} {ℓ'} {B : Type ℓ'} → is-hlevel B 4 → (f : A → B) → (fconst : ∀ a b → f a ≡ f b) → (fcoh : ∀ a b c → PathP (λ i → f a ≡ fconst b c i) (fconst a b) (fconst a c)) → (∀ a b c d → PathP (λ i → PathP (λ j → f a ≡ fcoh b c d i j) (fconst a b) (fcoh a c d i)) (fcoh a b c) (fcoh a b d)) → ∥ A ∥⁴ → B unquoteDef ∥-∥⁴-rec = make-rec-n 4 ∥-∥⁴-rec (quote ∥_∥⁴) ``` </details> <!-- ```agda open import Meta.Idiom open import Meta.Bind instance Map-∥∥ : Map (eff ∥_∥) Map-∥∥ .Map.map = ∥-∥-map Idiom-∥∥ : Idiom (eff ∥_∥) Idiom-∥∥ .Idiom.pure = inc Idiom-∥∥ .Idiom._<*>_ {A = A} {B = B} = go where go : ∥ (A → B) ∥ → ∥ A ∥ → ∥ B ∥ go (inc f) (inc x) = inc (f x) go (inc f) (squash x y i) = squash (go (inc f) x) (go (inc f) y) i go (squash f g i) y = squash (go f y) (go g y) i Bind-∥∥ : Bind (eff ∥_∥) Bind-∥∥ .Bind._>>=_ {A = A} {B = B} = go where go : ∥ A ∥ → (A → ∥ B ∥) → ∥ B ∥ go (inc x) f = f x go (squash x y i) f = squash (go x f) (go y f) i ``` -->