De Morgan's laws for Pred

[Taneb]

Closes #2831

[JacquesCarette]

Two comments, two requests for change.

Massively surprised this wasn't here already!

[JacquesCarette]

(again, would curry' typecheck?_

[jamesmckinna]

So... I think that all of these things are already covered by Relation.Nullary.Negation (including even @JacquesCarette 's observations about curry etc.)? Or not!?

[JacquesCarette]

I think you are correct @jamesmckinna . But there must also be something very slightly different about them? Perhaps some subtlety about irrelevance?

[jamesmckinna]

The differences, AFAICT, are that the lemmas in Relation.Nullary.Negation only concern the Universal (explicitly quantified) cases, whereas this PR also covers the IUniversal (implicitly quantified) ones as well. But, FTR, using the suggested import:

open import Relation.Nullary.Negation using (¬_; ∃⟶¬∀¬; ∀⟶¬∃¬; ¬∃⟶∀¬; ∀¬⟶¬∃; ∃¬⟶¬∀)
module _ {P : Pred A ℓ} where

  ¬∃⟨P⟩⇒Π[∁P] : ¬ ∃⟨ P ⟩ → Π[ ∁ P ]
  ¬∃⟨P⟩⇒Π[∁P] = ¬∃⟶∀¬

  ¬∃⟨P⟩⇒∀[∁P] : ¬ ∃⟨ P ⟩ → ∀[ ∁ P ]
  ¬∃⟨P⟩⇒∀[∁P] ¬sat = ¬∃⟶∀¬ ¬sat _

  ∃⟨∁P⟩⇒¬Π[P] : ∃⟨ ∁ P ⟩ → ¬ Π[ P ]
  ∃⟨∁P⟩⇒¬Π[P] = ∃¬⟶¬∀

  ∃⟨∁P⟩⇒¬∀[P] : ∃⟨ ∁ P ⟩ → ¬ ∀[ P ]
  ∃⟨∁P⟩⇒¬∀[P] ∃CP ∀P = ∃¬⟶¬∀ ∃CP λ _ → ∀P

  Π[∁P]⇒¬∃[P] : Π[ ∁ P ] → ¬ ∃⟨ P ⟩
  Π[∁P]⇒¬∃[P] = ∀¬⟶¬∃

  ∀[∁P]⇒¬∃[P] : ∀[ ∁ P ] → ¬ ∃⟨ P ⟩
  ∀[∁P]⇒¬∃[P] ∀∁P = ∀¬⟶¬∃ λ _ → ∀∁P

So... I think this is worth badging as status:duplicate or even status:invalid?

[jamesmckinna]

Do we want to merge this, or close, and follow up with a rename+deprecate as discussed in #2831 ?

I'm (largely) agnostic/uncommitted on this issue (apart from the redundancy involved), but proofs which differ at all from those versions offered above would be... redundancy too far, IMNSVHO ;-)