Using irrelevant fields

问题

Is it possible to declare fields in a record irrelevant but still use them somewhere?

Suppose I have

postulate f : ℕ → ℕ
record Silly x : Set where
  field
    n : ℕ
    s : f n ≡ x

open Silly

Then, I can

same-silly : ∀{x} {p q : Silly x} → f (n p) ≡ f(n q)
same-silly {x} {p} {q} = ≡-trans (s p) (≡-sym (s q))

But if I use dotted-fields,

record Silly x : Set where
  field
    n : ℕ
    .s : f n ≡ x  -- note the dot

then same-silly can no longer be proven ---not by me at least. When I attempt to use s, as in the definition above, Agda informs me Identifier s is declared irrelevant, so it cannot be used here.

I understand that I declared s irrelevant by dotting it, but I still want a bit of access to it: enough access to define same-silly. Is there a way to have this little bit but not complete access? What I mean is that I am thinking of s as a coherency condition that I'd like to use once in a blue-moon, but not terrible important and so the dot. Maybe ignore this paragraph.

Is it possible to form same-silly and have s be dotted?

(

Aside: does the dotted pattern mean propositional equality coincides with the relation

_~_ : ∀{x} (p q : Silly x) → Set
p ~ q = n p ≡ n q

? In particular, by adding to the record constructor _#_, I can show

eq : ∀{m x}{p q : f m ≡ x} → (m # p) ≡ (m # q) 
eq = ≡-refl

but I'm not yet certain that dotting fields means that definitionally equality ignores the dotted fields.

I'm browsing http://wiki.portal.chalmers.se/agda/agda.php?n=ForkedReferenceManual.Records#Irrelevantfields; which says that there are projections for dotted fields, but it seems I cannot use them anywhere.

)

Any help is appreciated!

回答1:

You can use irrelevant projections, but only when you are working in an irrelevant context. One way to enter an irrelevant context is to make the proof itself irrelevant:

.same-silly : ∀{x} {p q : Silly x} → f (n p) ≡ f(n q)
same-silly {x} {p} {q} = ≡-trans (s p) (≡-sym (s q))

This means that you will only be able to use the same-silly proof in irrelevant contexts as well.

Regarding your second question, the answer is yes: you can prove the following:

silly-equality : ∀ {n} {x y : Silly n} → x ~ y → x ≡ y
silly-equality refl = refl

So equality of the first projection (your ~ relation) really corresponds to propositional equality.

来源：https://stackoverflow.com/questions/34585992/using-irrelevant-fields

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