------------------------------------------------------------------------
-- Containers, including a definition of bag equivalence
------------------------------------------------------------------------
{-# OPTIONS --without-K #-}
module Container where
open import Bag-equivalence using (Kind); open Bag-equivalence.Kind
open import Equality.Propositional
open import Logical-equivalence hiding (id; _∘_; inverse)
open import Prelude hiding (id; List; map; lookup)
open import Bijection equality-with-J as Bijection
using (_↔_; module _↔_)
open import Equivalence equality-with-J as Eq
using (Is-equivalence; _≃_; ⟨_,_⟩; module _≃_)
open import Function-universe equality-with-J as Function-universe
hiding (inverse; Kind) renaming (_∘_ to _⟨∘⟩_)
open import H-level equality-with-J
open import H-level.Closure equality-with-J
open import Surjection equality-with-J using (module _↠_)
------------------------------------------------------------------------
-- Containers
record Container c : Set (lsuc c) where
constructor _▷_
field
Shape : Set c
Position : Shape → Set c
open Container public
-- Interpretation of containers.
⟦_⟧ : ∀ {c ℓ} → Container c → Set ℓ → Set _
⟦ S ▷ P ⟧ A = ∃ λ (s : S) → (P s → A)
------------------------------------------------------------------------
-- Some projections
-- The shape of something.
shape : ∀ {a c} {A : Set a} {C : Container c} → ⟦ C ⟧ A → Shape C
shape = proj₁
-- A lookup function.
lookup : ∀ {a c} {A : Set a} {C : Container c}
(xs : ⟦ C ⟧ A) → Position C (shape xs) → A
lookup = proj₂
------------------------------------------------------------------------
-- Map
-- Containers are functors.
map : ∀ {c x y} {C : Container c} {X : Set x} {Y : Set y} →
(X → Y) → ⟦ C ⟧ X → ⟦ C ⟧ Y
map f = Σ-map id (λ g → f ∘ g)
module Map where
identity : ∀ {c x} {C : Container c} {X : Set x}
(xs : ⟦ C ⟧ X) → map id xs ≡ xs
identity xs = refl
composition : ∀ {c x y z}
{C : Container c} {X : Set x} {Y : Set y} {Z : Set z}
(f : Y → Z) (g : X → Y) (xs : ⟦ C ⟧ X) →
map f (map g xs) ≡ map (f ∘ g) xs
composition f g xs = refl
-- Naturality.
Natural : ∀ {c₁ c₂ a} {C₁ : Container c₁} {C₂ : Container c₂} →
({A : Set a} → ⟦ C₁ ⟧ A → ⟦ C₂ ⟧ A) → Set (c₁ ⊔ c₂ ⊔ lsuc a)
Natural function =
∀ {A B} (f : A → B) xs →
map f (function xs) ≡ function (map f xs)
-- Natural transformations.
infixr 4 _[_]⟶_
record _[_]⟶_ {c₁ c₂} (C₁ : Container c₁) ℓ (C₂ : Container c₂) :
Set (c₁ ⊔ c₂ ⊔ lsuc ℓ) where
field
function : {A : Set ℓ} → ⟦ C₁ ⟧ A → ⟦ C₂ ⟧ A
natural : Natural function
-- Natural isomorphisms.
record _[_]↔_ {c₁ c₂} (C₁ : Container c₁) ℓ (C₂ : Container c₂) :
Set (c₁ ⊔ c₂ ⊔ lsuc ℓ) where
field
isomorphism : {A : Set ℓ} → ⟦ C₁ ⟧ A ↔ ⟦ C₂ ⟧ A
natural : Natural (_↔_.to isomorphism)
-- Natural isomorphisms are natural transformations.
natural-transformation : C₁ [ ℓ ]⟶ C₂
natural-transformation = record
{ function = _↔_.to isomorphism
; natural = natural
}
-- Natural isomorphisms can be inverted.
inverse : C₂ [ ℓ ]↔ C₁
inverse = record
{ isomorphism = Function-universe.inverse isomorphism
; natural = λ f xs →
map f (from xs) ≡⟨ sym $ left-inverse-of _ ⟩
from (to (map f (from xs))) ≡⟨ sym $ cong from $ natural f (from xs) ⟩
from (map f (to (from xs))) ≡⟨ cong (from ∘ map f) $ right-inverse-of _ ⟩∎
from (map f xs) ∎
}
where open module I {A : Set ℓ} = _↔_ (isomorphism {A = A})
open Function-universe using (inverse)
------------------------------------------------------------------------
-- Any, _∈_, bag equivalence and similar relations
-- Definition of Any for containers.
Any : ∀ {a c p} {A : Set a} {C : Container c} →
(A → Set p) → (⟦ C ⟧ A → Set (c ⊔ p))
Any {C = S ▷ P} Q (s , f) = ∃ λ (p : P s) → Q (f p)
-- Membership predicate.
infix 4 _∈_
_∈_ : ∀ {a c} {A : Set a} {C : Container c} → A → ⟦ C ⟧ A → Set _
x ∈ xs = Any (λ y → x ≡ y) xs
-- Bag equivalence etc. Note that the containers can be different as
-- long as the elements they contain have equal types.
infix 4 _∼[_]_
_∼[_]_ : ∀ {a c₁ c₂}
{A : Set a} {C₁ : Container c₁} {C₂ : Container c₂} →
⟦ C₁ ⟧ A → Kind → ⟦ C₂ ⟧ A → Set _
xs ∼[ k ] ys = ∀ z → z ∈ xs ↝[ k ] z ∈ ys
-- Bag equivalence.
infix 4 _≈-bag_
_≈-bag_ : ∀ {a c₁ c₂}
{A : Set a} {C₁ : Container c₁} {C₂ : Container c₂} →
⟦ C₁ ⟧ A → ⟦ C₂ ⟧ A → Set _
xs ≈-bag ys = xs ∼[ bag ] ys
------------------------------------------------------------------------
-- Various properties related to Any, _∈_ and _∼[_]_
-- Lemma relating Any to map.
Any-map : ∀ {a b c p} {A : Set a} {B : Set b} {C : Container c}
(P : B → Set p) (f : A → B) (xs : ⟦ C ⟧ A) →
Any P (map f xs) ↔ Any (P ∘ f) xs
Any-map P f xs = Any P (map f xs) □
-- Any can be expressed using _∈_.
Any-∈ : ∀ {a c p} {A : Set a} {C : Container c}
(P : A → Set p) (xs : ⟦ C ⟧ A) →
Any P xs ↔ ∃ λ x → P x × x ∈ xs
Any-∈ P (s , f) =
(∃ λ p → P (f p)) ↔⟨ ∃-cong (λ p → ∃-intro P (f p)) ⟩
(∃ λ p → ∃ λ x → P x × x ≡ f p) ↔⟨ ∃-comm ⟩
(∃ λ x → ∃ λ p → P x × x ≡ f p) ↔⟨ ∃-cong (λ _ → ∃-comm) ⟩
(∃ λ x → P x × ∃ λ p → x ≡ f p) □
-- Using this property we can prove that Any and _⊎_ commute.
Any-⊎ : ∀ {a c p q} {A : Set a} {C : Container c}
(P : A → Set p) (Q : A → Set q) (xs : ⟦ C ⟧ A) →
Any (λ x → P x ⊎ Q x) xs ↔ Any P xs ⊎ Any Q xs
Any-⊎ P Q xs =
Any (λ x → P x ⊎ Q x) xs ↔⟨ Any-∈ (λ x → P x ⊎ Q x) xs ⟩
(∃ λ x → (P x ⊎ Q x) × x ∈ xs) ↔⟨ ∃-cong (λ x → ×-⊎-distrib-right) ⟩
(∃ λ x → P x × x ∈ xs ⊎ Q x × x ∈ xs) ↔⟨ ∃-⊎-distrib-left ⟩
(∃ λ x → P x × x ∈ xs) ⊎ (∃ λ x → Q x × x ∈ xs) ↔⟨ inverse $ Any-∈ P xs ⊎-cong Any-∈ Q xs ⟩
Any P xs ⊎ Any Q xs □
-- Any preserves functions of various kinds and respects bag
-- equivalence and similar relations.
Any-cong : ∀ {k a c d p q}
{A : Set a} {C : Container c} {D : Container d}
(P : A → Set p) (Q : A → Set q)
(xs : ⟦ C ⟧ A) (ys : ⟦ D ⟧ A) →
(∀ x → P x ↝[ k ] Q x) → xs ∼[ k ] ys →
Any P xs ↝[ k ] Any Q ys
Any-cong P Q xs ys P↔Q xs∼ys =
Any P xs ↔⟨ Any-∈ P xs ⟩
(∃ λ z → P z × z ∈ xs) ↝⟨ ∃-cong (λ z → P↔Q z ×-cong xs∼ys z) ⟩
(∃ λ z → Q z × z ∈ ys) ↔⟨ inverse (Any-∈ Q ys) ⟩
Any Q ys □
-- Map preserves the relations.
map-cong : ∀ {k a b c d}
{A : Set a} {B : Set b} {C : Container c} {D : Container d}
(f : A → B)
(xs : ⟦ C ⟧ A) (ys : ⟦ D ⟧ A) →
xs ∼[ k ] ys → map f xs ∼[ k ] map f ys
map-cong f xs ys xs∼ys = λ z →
z ∈ map f xs ↔⟨ Any-map (_≡_ z) f xs ⟩
Any (λ x → z ≡ f x) xs ↝⟨ Any-cong _ _ xs ys (λ x → z ≡ f x □) xs∼ys ⟩
Any (λ x → z ≡ f x) ys ↔⟨ inverse (Any-map (_≡_ z) f ys) ⟩
z ∈ map f ys □
-- Lemma relating Any to if_then_else_.
Any-if : ∀ {a c p} {A : Set a} {C : Container c}
(P : A → Set p) (xs ys : ⟦ C ⟧ A) b →
Any P (if b then xs else ys) ↔
T b × Any P xs ⊎ T (not b) × Any P ys
Any-if P xs ys =
inverse ∘ if-lemma (λ b → Any P (if b then xs else ys)) id id
-- One can reconstruct (up to natural isomorphism) the shape set and
-- the position predicate from the interpretation and the Any
-- predicate transformer.
--
-- (The following lemmas were suggested by an anonymous reviewer.)
Shape′ : ∀ {c} → (Set → Set c) → Set c
Shape′ F = F ⊤
Shape-⟦⟧ : ∀ {c} (C : Container c) →
Shape C ↔ Shape′ ⟦ C ⟧
Shape-⟦⟧ C =
Shape C ↔⟨ inverse ×-right-identity ⟩
Shape C × ⊤ ↔⟨ ∃-cong (λ _ → inverse →-right-zero) ⟩
(∃ λ (s : Shape C) → Position C s → ⊤) □
Position′ : ∀ {c} (F : Set → Set c) →
({A : Set} → (A → Set) → (F A → Set c)) →
Shape′ F → Set c
Position′ _ Any = Any (λ (_ : ⊤) → ⊤)
Position-Any : ∀ {c} {C : Container c} (s : Shape C) →
Position C s ↔
Position′ ⟦ C ⟧ Any (_↔_.to (Shape-⟦⟧ C) s)
Position-Any {C = C} s =
Position C s ↔⟨ inverse ×-right-identity ⟩
Position C s × ⊤ □
expressed-in-terms-of-interpretation-and-Any :
∀ {c ℓ} (C : Container c) →
C [ ℓ ]↔ (⟦ C ⟧ ⊤ ▷ Any (λ _ → ⊤))
expressed-in-terms-of-interpretation-and-Any C = record
{ isomorphism = λ {A} →
(∃ λ (s : Shape C) → Position C s → A) ↔⟨ Σ-cong (Shape-⟦⟧ C) (λ _ → lemma) ⟩
(∃ λ (s : Shape′ ⟦ C ⟧) → Position′ ⟦ C ⟧ Any s → A) □
; natural = λ _ _ → refl
}
where
-- If equality of functions had been extensional, then the following
-- lemma could have been replaced by a congruence lemma applied to
-- Position-Any.
lemma : ∀ {a b} {A : Set a} {B : Set b} → (B → A) ↔ (B × ⊤ → A)
lemma = record
{ surjection = record
{ logical-equivalence = record
{ to = λ { f (p , tt) → f p }
; from = λ f p → f (p , tt)
}
; right-inverse-of = λ _ → refl
}
; left-inverse-of = λ _ → refl
}
------------------------------------------------------------------------
-- Alternative definition of bag equivalence
-- Two things are bag equal if there is a bijection (or equivalence)
-- between their positions which relates equal things.
infix 4 _≈[_]′_
_≈[_]′_ : ∀ {a c d} {A : Set a} {C : Container c} {D : Container d} →
⟦ C ⟧ A → Isomorphism-kind → ⟦ D ⟧ A → Set _
_≈[_]′_ {C = C} {D} (s , f) k (s′ , f′) =
∃ λ (P↔P : Position C s ↔[ k ] Position D s′) →
(∀ p → f p ≡ f′ (to-implication P↔P p))
-- If the position sets are sets (have H-level two), then the two
-- instantiations of _≈[_]′_ are isomorphic (assuming extensionality).
≈′↔≈′ : ∀ {a c d} {A : Set a} {C : Container c} {D : Container d} →
Extensionality (c ⊔ d) (c ⊔ d) →
(∀ s → Is-set (Position C s)) →
(xs : ⟦ C ⟧ A) (ys : ⟦ D ⟧ A) →
xs ≈[ bag ]′ ys ↔ xs ≈[ bag-with-equivalence ]′ ys
≈′↔≈′ ext P-set (s , f) (s′ , f′) =
(∃ λ P↔P → ∀ p → f p ≡ f′ (to-implication P↔P p)) ↔⟨ Σ-cong (Eq.↔↔≃ ext (P-set s)) (λ _ → Bijection.id) ⟩
(∃ λ P↔P → ∀ p → f p ≡ f′ (to-implication P↔P p)) □
-- The definition _≈[_]′_ is also logically equivalent to the one
-- given above. The proof is very similar to the one given in
-- Bag-equivalence.
-- Membership can be expressed as "there is an index which points to
-- the element". In fact, membership /is/ expressed in this way, so
-- this proof is unnecessary.
∈-lookup : ∀ {a c} {A : Set a} {C : Container c} {z}
(xs : ⟦ C ⟧ A) → z ∈ xs ↔ ∃ λ p → z ≡ lookup xs p
∈-lookup {z = z} xs = z ∈ xs □
-- The index which points to the element (not used below).
index : ∀ {a c} {A : Set a} {C : Container c} {z}
(xs : ⟦ C ⟧ A) → z ∈ xs → Position C (shape xs)
index xs = proj₁ ∘ to-implication (∈-lookup xs)
-- The positions for a given shape can be expressed in terms of the
-- membership predicate.
Position-shape : ∀ {a c} {A : Set a} {C : Container c} (xs : ⟦ C ⟧ A) →
(∃ λ z → z ∈ xs) ↔ Position C (shape xs)
Position-shape {C = C} (s , f) =
(∃ λ z → ∃ λ p → z ≡ f p) ↔⟨ ∃-comm ⟩
(∃ λ p → ∃ λ z → z ≡ f p) ↔⟨⟩
(∃ λ p → Singleton (f p)) ↔⟨ ∃-cong (λ _ → inverse (_⇔_.to contractible⇔⊤↔ (singleton-contractible _))) ⟩
Position C s × ⊤ ↔⟨ ×-right-identity ⟩
Position C s □
-- Position _ ∘ shape respects the various relations.
Position-shape-cong :
∀ {k a c d} {A : Set a} {C : Container c} {D : Container d}
(xs : ⟦ C ⟧ A) (ys : ⟦ D ⟧ A) →
xs ∼[ k ] ys → Position C (shape xs) ↝[ k ] Position D (shape ys)
Position-shape-cong {C = C} {D} xs ys xs∼ys =
Position C (shape xs) ↔⟨ inverse $ Position-shape xs ⟩
∃ (λ z → z ∈ xs) ↝⟨ ∃-cong xs∼ys ⟩
∃ (λ z → z ∈ ys) ↔⟨ Position-shape ys ⟩
Position D (shape ys) □
-- Furthermore Position-shape-cong relates equal elements.
Position-shape-cong-relates :
∀ {k a c d} {A : Set a} {C : Container c} {D : Container d}
(xs : ⟦ C ⟧ A) (ys : ⟦ D ⟧ A) (xs≈ys : xs ∼[ k ] ys) p →
lookup xs p ≡
lookup ys (to-implication (Position-shape-cong xs ys xs≈ys) p)
Position-shape-cong-relates {bag} xs ys xs≈ys p =
lookup xs p ≡⟨ proj₂ $ to-implication (xs≈ys (lookup xs p)) (p , refl) ⟩
lookup ys (proj₁ $ to-implication (xs≈ys (lookup xs p))
(p , refl)) ≡⟨ refl ⟩
lookup ys (_↔_.to (Position-shape ys) $
Σ-map id (λ {z} → to-implication (xs≈ys z)) $
_↔_.from (Position-shape xs) $ p) ≡⟨ refl ⟩
lookup ys (_↔_.to (Position-shape ys) $
to-implication (∃-cong xs≈ys) $
_↔_.from (Position-shape xs) $ p) ≡⟨ refl ⟩
lookup ys (to-implication
((from-bijection (Position-shape ys) ⟨∘⟩
∃-cong xs≈ys) ⟨∘⟩
from-bijection (inverse $ Position-shape xs))
p) ≡⟨ refl ⟩∎
lookup ys (to-implication (Position-shape-cong xs ys xs≈ys) p) ∎
Position-shape-cong-relates {bag-with-equivalence} xs ys xs≈ys p =
proj₂ $ to-implication (xs≈ys (lookup xs p)) (p , refl)
Position-shape-cong-relates {subbag} xs ys xs≈ys p =
proj₂ $ to-implication (xs≈ys (lookup xs p)) (p , refl)
Position-shape-cong-relates {set} xs ys xs≈ys p =
proj₂ $ to-implication (xs≈ys (lookup xs p)) (p , refl)
Position-shape-cong-relates {subset} xs ys xs≈ys p =
proj₂ $ to-implication (xs≈ys (lookup xs p)) (p , refl)
Position-shape-cong-relates {surjection} xs ys xs≈ys p =
proj₂ $ to-implication (xs≈ys (lookup xs p)) (p , refl)
-- We get that the two definitions of bag equivalence are logically
-- equivalent.
≈⇔≈′ : ∀ {k a c d} {A : Set a} {C : Container c} {D : Container d}
(xs : ⟦ C ⟧ A) (ys : ⟦ D ⟧ A) →
xs ∼[ ⌊ k ⌋-iso ] ys ⇔ xs ≈[ k ]′ ys
≈⇔≈′ {k} xs ys = record
{ to = λ xs≈ys → ( Position-shape-cong xs ys xs≈ys
, Position-shape-cong-relates xs ys xs≈ys
)
; from = from
}
where
from : xs ≈[ k ]′ ys → xs ∼[ ⌊ k ⌋-iso ] ys
from (P↔P , related) = λ z →
z ∈ xs ↔⟨⟩
∃ (λ p → z ≡ lookup xs p) ↔⟨ Σ-cong P↔P (λ p → _↠_.from (Π≡↔≡-↠-≡ k _ _) (related p) z) ⟩
∃ (λ p → z ≡ lookup ys p) ↔⟨⟩
z ∈ ys □
-- If equivalences are used, then the definitions are isomorphic
-- (assuming extensionality).
--
-- Thierry Coquand helped me with this proof: At first I wasn't sure
-- if it was true or not, but then I managed to prove it for singleton
-- lists, Thierry found a proof for lists of length two, I found one
-- for streams, and finally I could complete the proof below.
≈↔≈′ : ∀ {a c d} {A : Set a} {C : Container c} {D : Container d} →
Extensionality (a ⊔ c ⊔ d) (a ⊔ c ⊔ d) →
(xs : ⟦ C ⟧ A) (ys : ⟦ D ⟧ A) →
xs ∼[ bag-with-equivalence ] ys ↔
xs ≈[ bag-with-equivalence ]′ ys
≈↔≈′ {a} {c} {d} {C = C} {D} ext xs ys = record
{ surjection = record
{ logical-equivalence = equiv
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
equiv = ≈⇔≈′ {k = equivalence} xs ys
open _⇔_ equiv
from∘to : ∀ xs≈ys → from (to xs≈ys) ≡ xs≈ys
from∘to xs≈ys =
lower-extensionality (c ⊔ d) a ext λ z →
Eq.lift-equality ext $
lower-extensionality d c ext λ { (p , z≡xs[p]) →
let xs[p]≡ys[-] : ∃ λ p′ → lookup xs p ≡ lookup ys p′
xs[p]≡ys[-] = _≃_.to (xs≈ys (lookup xs p)) (p , refl) in
Σ-map id (trans z≡xs[p]) xs[p]≡ys[-] ≡⟨ elim₁ (λ {z} z≡xs[p] → Σ-map id (trans z≡xs[p]) xs[p]≡ys[-] ≡
_≃_.to (xs≈ys z) (p , z≡xs[p]))
(Σ-map id (trans refl) xs[p]≡ys[-] ≡⟨ cong (_,_ _) (trans-reflˡ _) ⟩∎
xs[p]≡ys[-] ∎)
z≡xs[p] ⟩∎
_≃_.to (xs≈ys z) (p , z≡xs[p]) ∎ }
to∘from : ∀ xs≈ys → to (from xs≈ys) ≡ xs≈ys
to∘from (⟨ f , f-eq ⟩ , related) = Σ-≡,≡→≡ f≡f
(subst (P ∘ _≃_.to) f≡f (trans refl ∘ related) ≡⟨ cong (subst (P ∘ _≃_.to) f≡f)
(lower-extensionality (a ⊔ d) (c ⊔ d) ext λ _ → trans-reflˡ _) ⟩
subst (P ∘ _≃_.to) f≡f related ≡⟨ subst-∘ P _≃_.to f≡f ⟩
subst P (cong _≃_.to f≡f) related ≡⟨ cong (λ eq → subst P eq related) cong-to-f≡f ⟩
subst P refl related ≡⟨ subst-refl P {x = f} related ⟩∎
related ∎)
where
P : (Position C (shape xs) → Position D (shape ys)) → Set (a ⊔ c)
P f = ∀ p → lookup xs p ≡ lookup ys (f p)
f-eq′ : Is-equivalence f
f-eq′ = _≃_.is-equivalence $
Position-shape-cong xs ys $
from (⟨ f , f-eq ⟩ , related)
abstract
irr : f-eq′ ≡ f-eq
irr = proj₁ $
Eq.propositional (lower-extensionality a a ext) f _ _
f≡f : ⟨ f , f-eq′ ⟩ ≡ ⟨ f , f-eq ⟩
f≡f = cong (⟨_,_⟩ f) irr
cong-to-f≡f : cong _≃_.to f≡f ≡ refl {x = f}
cong-to-f≡f =
cong _≃_.to f≡f ≡⟨ cong-∘ _≃_.to (⟨_,_⟩ f) irr ⟩
cong (_≃_.to ∘ ⟨_,_⟩ f) irr ≡⟨ cong-const irr ⟩∎
refl ∎
------------------------------------------------------------------------
-- Another alternative definition of bag equivalence
-- A higher-order variant of _∼[_]_. Note that this definition is
-- large (due to the quantification over predicates).
infix 4 _∼[_]″_
_∼[_]″_ : ∀ {a c d} {A : Set a} {C : Container c} {D : Container d} →
⟦ C ⟧ A → Kind → ⟦ D ⟧ A → Set (lsuc a ⊔ c ⊔ d)
_∼[_]″_ {a} {A = A} xs k ys =
(P : A → Set a) → Any P xs ↝[ k ] Any P ys
-- This definition is logically equivalent to _∼[_]_.
∼⇔∼″ : ∀ {k a c d} {A : Set a} {C : Container c} {D : Container d}
(xs : ⟦ C ⟧ A) (ys : ⟦ D ⟧ A) →
xs ∼[ k ] ys ⇔ xs ∼[ k ]″ ys
∼⇔∼″ xs ys = record
{ to = λ xs∼ys P → Any-cong P P xs ys (λ _ → id) xs∼ys
; from = λ Any-xs↝Any-ys z → Any-xs↝Any-ys (λ x → z ≡ x)
}