Tom de Jong, 24 January 2022
(Based on code from FreeJoinSemiLattice.lagda written 18-24 December 2020.)
TODO. Comment.
\begin{code}
{-# OPTIONS --safe --without-K #-}
open import MLTT.Spartan
open import UF.PropTrunc
module UF.Powerset-Fin
(pt : propositional-truncations-exist)
where
open PropositionalTruncation pt
open import Fin.ArithmeticViaEquivalence
open import Fin.Type
open import Fin.Kuratowski pt
open import MLTT.List
open import Notation.UnderlyingType
open import OrderedTypes.JoinSemiLattices
open import UF.Base
open import UF.Equiv
open import UF.EquivalenceExamples
open import UF.FunExt
open import UF.Lower-FunExt
open import UF.ImageAndSurjection pt
open import UF.Powerset
open import UF.Sets
open import UF.Sets-Properties
open import UF.Subsingletons
open binary-unions-of-subsets pt
is-Kuratowski-finite-subset : {X : 𝓤 ̇ } (A : 𝓟 X) → 𝓤 ̇
is-Kuratowski-finite-subset A = is-Kuratowski-finite (𝕋 A)
∅-is-Kuratowski-finite-subset : {X : 𝓤 ̇ }
→ is-Kuratowski-finite-subset ∅
∅-is-Kuratowski-finite-subset {𝓤} {X} = ∣ 0 , e , σ ∣
where
e : Fin 0 → 𝕋 {𝓤} {X} ∅
e = 𝟘-elim
σ : is-surjection e
σ (x , x-in-emptyset) = 𝟘-elim x-in-emptyset
module _
{X : 𝓤 ̇ }
(X-is-set : is-set X)
where
open singleton-subsets X-is-set
❴❵-is-Kuratowski-finite-subset : {x : X}
→ is-Kuratowski-finite-subset ❴ x ❵
❴❵-is-Kuratowski-finite-subset {x} = ∣ 1 , e , σ ∣
where
e : Fin 1 → 𝕋 ❴ x ❵
e 𝟎 = x , refl
σ : is-surjection e
σ (x , refl) = ∣ inr ⋆ , refl ∣
\end{code}
We proceed to show that Kuratowski finite subsets are closed under binary unions.
\begin{code}
module _
{X : 𝓤 ̇ }
where
∪-enum' : (A B : 𝓟 X) {n m : ℕ}
→ (Fin n → 𝕋 A)
→ (Fin m → 𝕋 B)
→ (Fin n + Fin m) → 𝕋 (A ∪ B)
∪-enum' A B e f (inl k) = (𝕋-to-carrier A (e k) ,
∪-is-upperbound₁ A B (𝕋-to-carrier A (e k))
(𝕋-to-membership A (e k)))
∪-enum' A B e f (inr k) = (𝕋-to-carrier B (f k) ,
∪-is-upperbound₂ A B (𝕋-to-carrier B (f k))
(𝕋-to-membership B (f k)))
∪-enum : (A B : 𝓟 X) {n m : ℕ}
→ (Fin n → 𝕋 A)
→ (Fin m → 𝕋 B)
→ Fin (n +' m) → 𝕋 (A ∪ B)
∪-enum A B {n} {m} e f = ∪-enum' A B e f ∘ ⌜ Fin+homo n m ⌝
∪-enum'-is-surjection : (A B : 𝓟 X) {n m : ℕ}
(e : Fin n → 𝕋 A)
(f : Fin m → 𝕋 B)
→ is-surjection e
→ is-surjection f
→ is-surjection (∪-enum' A B e f)
∪-enum'-is-surjection A B {n} {m} e f σ τ (x , p) = ∥∥-rec ∥∥-is-prop γ p
where
γ : (x ∈ A + x ∈ B)
→ ∃ c ꞉ (Fin n + Fin m) , ∪-enum' A B e f c ＝ (x , p)
γ (inl a) = ∥∥-functor γ₁ (σ (x , a))
where
γ₁ : (Σ k ꞉ Fin n , e k ＝ (x , a))
→ Σ c ꞉ (Fin n + Fin m) , ∪-enum' A B e f c ＝ (x , p)
γ₁ (k , p) = inl k , to-subtype-＝ (∈-is-prop (A ∪ B)) (ap pr₁ p)
γ (inr b) = ∥∥-functor γ₂ (τ (x , b))
where
γ₂ : (Σ k ꞉ Fin m , f k ＝ (x , b))
→ Σ c ꞉ (Fin n + Fin m) , ∪-enum' A B e f c ＝ (x , p)
γ₂ (k , p) = inr k , to-subtype-＝ (∈-is-prop (A ∪ B)) (ap pr₁ p)
∪-enum-is-surjection : (A B : 𝓟 X) {n m : ℕ}
(e : Fin n → 𝕋 A)
(f : Fin m → 𝕋 B)
→ is-surjection e
→ is-surjection f
→ is-surjection (∪-enum A B e f)
∪-enum-is-surjection A B {n} {m} e f σ τ =
∘-is-surjection
(equivs-are-surjections (⌜⌝-is-equiv (Fin+homo n m)))
(∪-enum'-is-surjection A B e f σ τ)
∪-is-Kuratowski-finite-subset : (A B : 𝓟 X)
→ is-Kuratowski-finite-subset A
→ is-Kuratowski-finite-subset B
→ is-Kuratowski-finite-subset (A ∪ B)
∪-is-Kuratowski-finite-subset A B kᴬ kᴮ = k
where
k : is-Kuratowski-finite-subset (A ∪ B)
k = ∥∥-functor₂ γ kᴬ kᴮ
where
γ : (Σ nᴬ ꞉ ℕ , Fin nᴬ ↠ 𝕋 A)
→ (Σ nᴮ ꞉ ℕ , Fin nᴮ ↠ 𝕋 B)
→ Σ n ꞉ ℕ , Fin n ↠ 𝕋 (A ∪ B)
γ (nᴬ , eᴬ , eᴬ-is-surj) (nᴮ , eᴮ , eᴮ-is-surj) =
(nᴬ +' nᴮ , ∪-enum A B eᴬ eᴮ
, ∪-enum-is-surjection A B eᴬ eᴮ eᴬ-is-surj eᴮ-is-surj)
\end{code}
The Kuratowski finite subsets (ordered by subset inclusion) are a natural
example of a join-semilattice, which we are going to prove now.
(In fact, the Kuratowski finite subsets are the free join-semilattice, see
FreeJoinSemiLattice.lagda.)
\begin{code}
𝓚 : (X : 𝓤 ̇ ) → 𝓤 ⁺ ̇
𝓚 X = Σ A ꞉ 𝓟 X , is-Kuratowski-finite-subset A
module _
{X : 𝓤 ̇ }
where
instance
underlying-type-of-𝓚 : Underlying-Type (𝓚 X) (𝓟 X)
⟨_⟩ {{underlying-type-of-𝓚}} (A , _) = A
⟨_⟩₂ : (A : 𝓚 X) → is-Kuratowski-finite-subset ⟨ A ⟩
⟨_⟩₂ = pr₂
_⊆[𝓚]_ : 𝓚 X → 𝓚 X → 𝓤 ̇
A ⊆[𝓚] B = ⟨ A ⟩ ⊆ ⟨ B ⟩
⊆[𝓚]-is-reflexive : (A : 𝓚 X) → A ⊆[𝓚] A
⊆[𝓚]-is-reflexive A = ⊆-refl ⟨ A ⟩
⊆[𝓚]-is-transitive : (A B C : 𝓚 X) → A ⊆[𝓚] B → B ⊆[𝓚] C → A ⊆[𝓚] C
⊆[𝓚]-is-transitive A B C = ⊆-trans ⟨ A ⟩ ⟨ B ⟩ ⟨ C ⟩
module singleton-Kuratowski-finite-subsets
(X-is-set : is-set X)
where
open singleton-subsets X-is-set
❴_❵[𝓚] : X → 𝓚 X
❴ x ❵[𝓚] = (❴ x ❵ , ❴❵-is-Kuratowski-finite-subset X-is-set)
∅[𝓚] : 𝓚 X
∅[𝓚] = ∅ , ∅-is-Kuratowski-finite-subset
∅[𝓚]-is-least : (A : 𝓚 X) → ∅[𝓚] ⊆[𝓚] A
∅[𝓚]-is-least A = ∅-is-least ⟨ A ⟩
_∪[𝓚]_ : 𝓚 X → 𝓚 X → 𝓚 X
_∪[𝓚]_ (A , k₁) (B , k₂) = (A ∪ B) , k
where
k : is-Kuratowski-finite-subset (A ∪ B)
k = ∪-is-Kuratowski-finite-subset A B k₁ k₂
∪[𝓚]-is-upperbound₁ : (A B : 𝓚 X) → A ⊆[𝓚] (A ∪[𝓚] B)
∪[𝓚]-is-upperbound₁ A B = ∪-is-upperbound₁ ⟨ A ⟩ ⟨ B ⟩
∪[𝓚]-is-upperbound₂ : (A B : 𝓚 X) → B ⊆[𝓚] (A ∪[𝓚] B)
∪[𝓚]-is-upperbound₂ A B = ∪-is-upperbound₂ ⟨ A ⟩ ⟨ B ⟩
∪[𝓚]-is-lowerbound-of-upperbounds : (A B C : 𝓚 X)
→ A ⊆[𝓚] C → B ⊆[𝓚] C → (A ∪[𝓚] B) ⊆[𝓚] C
∪[𝓚]-is-lowerbound-of-upperbounds A B C =
∪-is-lowerbound-of-upperbounds ⟨ A ⟩ ⟨ B ⟩ ⟨ C ⟩
module _
(fe : funext 𝓤 (𝓤 ⁺))
where
⊆[𝓚]-is-prop-valued : (A B : 𝓚 X) → is-prop (A ⊆[𝓚] B)
⊆[𝓚]-is-prop-valued A B = ⊆-is-prop (lower-funext 𝓤 (𝓤 ⁺) fe) ⟨ A ⟩ ⟨ B ⟩
module _
(pe : propext 𝓤)
where
⊆[𝓚]-is-antisymmetric : (A B : 𝓚 X) → A ⊆[𝓚] B → B ⊆[𝓚] A → A ＝ B
⊆[𝓚]-is-antisymmetric A B s t =
to-subtype-＝ (λ _ → being-Kuratowski-finite-is-prop)
(subset-extensionality pe fe s t)
𝓚-is-set : is-set (𝓚 X)
𝓚-is-set = subtypes-of-sets-are-sets' ⟨_⟩ s (powersets-are-sets fe pe)
where
s : left-cancellable ⟨_⟩
s e = to-subtype-＝ (λ _ → being-Kuratowski-finite-is-prop) e
\end{code}
We are now ready to prove that the Kuratowski finite subsets are a join-semilattice.
\begin{code}
module _
(pe : propext 𝓤)
(fe : funext 𝓤 (𝓤 ⁺))
(X : 𝓤 ̇ )
where
𝓚-join-semilattice : JoinSemiLattice (𝓤 ⁺) 𝓤
𝓚-join-semilattice = record {
L = 𝓚 X ;
L-is-set = 𝓚-is-set fe pe;
_⊑_ = _⊆[𝓚]_;
⊑-is-prop-valued = ⊆[𝓚]-is-prop-valued fe;
⊑-is-reflexive = ⊆[𝓚]-is-reflexive;
⊑-is-transitive = ⊆[𝓚]-is-transitive;
⊑-is-antisymmetric = ⊆[𝓚]-is-antisymmetric fe pe;
⊥ = ∅[𝓚];
⊥-is-least = ∅[𝓚]-is-least;
_∨_ = _∪[𝓚]_;
∨-is-upperbound₁ = ∪[𝓚]-is-upperbound₁;
∨-is-upperbound₂ = ∪[𝓚]-is-upperbound₂;
∨-is-lowerbound-of-upperbounds = ∪[𝓚]-is-lowerbound-of-upperbounds
}
\end{code}
Now that we have that the Kuratowski finite subsets are a join-semilattice, we
automatically have binary joins available, which will come in useful when
proving a general induction principle for Kuratowski finite subsets.
\begin{code}
module _
(X-is-set : is-set X)
where
open JoinSemiLattice 𝓚-join-semilattice
open singleton-Kuratowski-finite-subsets X-is-set
Kuratowski-finite-subset-expressed-as-finite-join :
(A : 𝓚 X)
{n : ℕ}
{e : Fin n → 𝕋 ⟨ A ⟩}
(σ : is-surjection e)
→ A ＝ ∨ⁿ (❴_❵[𝓚] ∘ 𝕋-to-carrier ⟨ A ⟩ ∘ e)
Kuratowski-finite-subset-expressed-as-finite-join A {n} {e} σ = γ
where
ε : Fin n → 𝓚 X
ε = ❴_❵[𝓚] ∘ 𝕋-to-carrier ⟨ A ⟩ ∘ e
γ : A ＝ ∨ⁿ ε
γ = ⊆[𝓚]-is-antisymmetric fe pe A (∨ⁿ ε) u v
where
u : A ⊆[𝓚] ∨ⁿ ε
u x a = ∥∥-rec (∈-is-prop ⟨ ∨ⁿ ε ⟩ x) μ (σ (x , a))
where
μ : (Σ k ꞉ Fin n , e k ＝ (x , a))
→ x ∈ ⟨ ∨ⁿ ε ⟩
μ (k , refl) = ∨ⁿ-is-upperbound ε k x refl
v : ∨ⁿ ε ⊆[𝓚] A
v = ∨ⁿ-is-lowerbound-of-upperbounds ε A ν
where
ν : (k : Fin n) → ε k ⊆[𝓚] A
ν k x refl = 𝕋-to-membership ⟨ A ⟩ (e k)
Kuratowski-finite-subset-induction :
(Q : 𝓚 X → 𝓣 ̇ )
→ ((A : 𝓚 X) → is-prop (Q A))
→ Q (∅[𝓚])
→ ((x : X) → Q (❴ x ❵[𝓚]))
→ ((A B : 𝓚 X) → Q A → Q B → Q (A ∪[𝓚] B))
→ (A : 𝓚 X) → Q A
Kuratowski-finite-subset-induction
Q Q-is-prop-valued Q-empty Q-singletons Q-unions 𝔸@(A , k) =
∥∥-rec (Q-is-prop-valued 𝔸) γ k
where
γ : (Σ n ꞉ ℕ , Fin n ↠ 𝕋 A) → Q 𝔸
γ (n , e , e-surj) = transport Q ϕ (ψ n (𝕋-to-carrier A ∘ e))
where
ϕ : ∨ⁿ (❴_❵[𝓚] ∘ 𝕋-to-carrier A ∘ e) ＝ 𝔸
ϕ = (Kuratowski-finite-subset-expressed-as-finite-join 𝔸 e-surj) ⁻¹
ψ : (m : ℕ) (f : Fin m → X) → Q (∨ⁿ (❴_❵[𝓚] ∘ f))
ψ zero f = Q-empty
ψ (succ m) f = Q-unions
(∨ⁿ (❴_❵[𝓚] ∘ f ∘ suc))
((❴_❵[𝓚] ∘ f) 𝟎)
IH
(Q-singletons (f 𝟎))
where
IH : Q (∨ⁿ (❴_❵[𝓚] ∘ f ∘ suc))
IH = ψ m (f ∘ suc)
\end{code}
We consider the canonical map from lists on X to the powerset of X and prove
that its image is exactly the type of Kuratowski finite powersets of X.
\begin{code}
module canonical-map-from-lists-to-subsets
{X : 𝓤 ̇ }
(X-is-set : is-set X)
where
open singleton-subsets X-is-set
open singleton-Kuratowski-finite-subsets X-is-set
κ : List X → 𝓟 X
κ [] = ∅
κ (x ∷ xs) = ❴ x ❵ ∪ κ xs
κ-of-concatenated-lists-is-union : propext 𝓤
→ funext 𝓤 (𝓤 ⁺)
→ (l l' : List X)
→ κ (l ++ l') ＝ κ l ∪ κ l'
κ-of-concatenated-lists-is-union pe fe [] l' =
∅-left-neutral-for-∪ pe fe (κ l') ⁻¹
κ-of-concatenated-lists-is-union pe fe (x ∷ l) l' =
❴ x ❵ ∪ κ (l ++ l') ＝⟨ ap (❴ x ❵ ∪_) IH ⟩
❴ x ❵ ∪ (κ l ∪ κ l') ＝⟨ (∪-assoc pe fe ❴ x ❵ (κ l) (κ l')) ⁻¹ ⟩
(❴ x ❵ ∪ κ l) ∪ κ l' ∎
where
IH : κ (l ++ l') ＝ (κ l ∪ κ l')
IH = κ-of-concatenated-lists-is-union pe fe l l'
κ-of-list-is-Kuratowski-finite-subset : (l : List X)
→ is-Kuratowski-finite-subset (κ l)
κ-of-list-is-Kuratowski-finite-subset [] = ∅-is-Kuratowski-finite-subset
κ-of-list-is-Kuratowski-finite-subset (x ∷ l) =
∪-is-Kuratowski-finite-subset ❴ x ❵ (κ l)
(❴❵-is-Kuratowski-finite-subset X-is-set)
(κ-of-list-is-Kuratowski-finite-subset l)
Kuratowski-finite-subset-if-in-image-of-κ : (A : 𝓟 X)
→ A ∈image κ
→ is-Kuratowski-finite-subset A
Kuratowski-finite-subset-if-in-image-of-κ A =
∥∥-rec being-Kuratowski-finite-is-prop γ
where
γ : (Σ l ꞉ List X , κ l ＝ A)
→ is-Kuratowski-finite-subset A
γ (l , refl) = κ-of-list-is-Kuratowski-finite-subset l
\end{code}
For proving the converse, the aforementioned induction principle for
Kuroratowski finite subsets comes in handy (as suggested by Martín Escardó
during an Agda Club meeting).
\begin{code}
in-image-of-κ-if-Kuratowski-finite-subset : propext 𝓤
→ funext 𝓤 (𝓤 ⁺)
→ (A : 𝓟 X)
→ is-Kuratowski-finite-subset A
→ A ∈image κ
in-image-of-κ-if-Kuratowski-finite-subset pe fe = goal
where
Q : 𝓚 X → 𝓤 ⁺ ̇
Q A = ⟨ A ⟩ ∈image κ
Q-is-prop-valued : (A : 𝓚 X) → is-prop (Q A)
Q-is-prop-valued A = being-in-the-image-is-prop ⟨ A ⟩ κ
Q-empty : Q ∅[𝓚]
Q-empty = ∣ [] , refl ∣
Q-singleton : (x : X) → Q ❴ x ❵[𝓚]
Q-singleton x = ∣ [ x ] , subset-extensionality pe fe s t ∣
where
s : κ [ x ] ⊆ ❴ x ❵
s y = ∥∥-rec (∈-is-prop ❴ x ❵ y) γ
where
γ : (y ∈ ❴ x ❵ + y ∈ κ []) → y ∈ ❴ x ❵
γ (inl p) = p
γ (inr q) = 𝟘-elim q
t : ❴ x ❵ ⊆ κ [ x ]
t y p = ∣ inl p ∣
Q-unions : (A B : 𝓚 X) → Q A → Q B → Q (A ∪[𝓚] B)
Q-unions A B qᴬ qᴮ = ∥∥-functor₂ γ qᴬ qᴮ
where
γ : (Σ lᴬ ꞉ List X , κ lᴬ ＝ ⟨ A ⟩)
→ (Σ lᴮ ꞉ List X , κ lᴮ ＝ ⟨ B ⟩)
→ (Σ l ꞉ List X , κ l ＝ ⟨ A ∪[𝓚] B ⟩)
γ (lᴬ , pᴬ) (lᴮ , pᴮ) = ((lᴬ ++ lᴮ) , p)
where
p = κ (lᴬ ++ lᴮ) ＝⟨ κ-of-concatenated-lists-is-union pe fe lᴬ lᴮ ⟩
κ lᴬ ∪ κ lᴮ ＝⟨ ap₂ _∪_ pᴬ pᴮ ⟩
⟨ A ⟩ ∪ ⟨ B ⟩ ∎
Q-holds-everywhere : (A : 𝓚 X) → Q A
Q-holds-everywhere = Kuratowski-finite-subset-induction pe fe X X-is-set
Q Q-is-prop-valued
Q-empty
Q-singleton
Q-unions
goal : (A : 𝓟 X) → is-Kuratowski-finite-subset A → A ∈image κ
goal A k = Q-holds-everywhere (A , k)
\end{code}
Putting this all together, we obtained the promised equivalence between the
image of κ and the Kuratowski finite subsets of X.
\begin{code}
image-of-κ-is-the-Kuratowski-finite-powerset : propext 𝓤
→ funext 𝓤 (𝓤 ⁺)
→ image κ ≃ 𝓚 X
image-of-κ-is-the-Kuratowski-finite-powerset pe fe = Σ-cong γ
where
γ : (A : 𝓟 X) → (A ∈image κ) ≃ is-Kuratowski-finite-subset A
γ A = logically-equivalent-props-are-equivalent
(being-in-the-image-is-prop A κ)
being-Kuratowski-finite-is-prop
(Kuratowski-finite-subset-if-in-image-of-κ A)
(in-image-of-κ-if-Kuratowski-finite-subset pe fe A)
\end{code}