Using the notion of subsets as predicates,
ℙ : ∀ {b a} → Set a → Set (a ⊔ suc b)
ℙ {b} {a} X = X → Set b
I'd like to consider structures endowed with a predicate on subsets,
record SetWithAPredicate {a c} : Set {!!} where
field
S : Set a
P : ∀ {b} → ℙ {b} S → Set c
This is an ill-formed construction due to the level quantification used in ℙ. Everything works fine when I use S, P as parameters to a module, but I'd like them to be records so that I can form constructions on them and give instances of them.
I've tried a few other things, such as moving the level b of ℙ inside the definition via an existential but then that led to metavaraible trouble. I also tried changing the type of P,
P : ℙ {a} S → Set c
but then I can no longer ask for, say, the empty set to have the property:
P-⊥ : P(λ _ → ⊥)
This is not well typed since Set != Set a ---I must admit, I tried to use Level.lift here, but failed to do so.
More generally, this will also not allow me to express closure properties, such as P is closed under arbitrary unions ---this is what I'm really interested in.
I understand that I can just avoid level polymorphism,
ℙ' : ∀ {a} → Set a → Set (suc zero ⊔ a)
ℙ' {a} X = X → Set
but then simple items such as the largest subset,
ℙ'-⊤ : ∀ {i} {A : Set i} → ℙ' A
ℙ'-⊤ {i} {A} = λ e → Σ a ∶ A • a ≡ e
-- Σ_∶_•_ is just syntax for Σ A (λ a → ...)
will not even typecheck!
Perhaps I did not realise the notion of subset as predicate appropriately ---any advice would be appreciated. Thank-you!