[ add/DRY ] Ex falso-type eliminator for irrelevant instances of irreflexive 'equality'

[jamesmckinna]

Having turned my focus to making certain relational arguments irrelevant in eg. Data.Fin.Base (#2783 #2790 cf. #2787 ), I observe a frequently-occurring pattern of reasoning, usually at the base case, where an 'irreflexive' argument of type 0 ≢ 0 ends up being eliminated via contradiction refl.

In order to make such arguments valid when such arguments are marked as irrelevant then requires such proofs to be modified, in-place, to use the new contradiction-irr introduced in #2785 .

This is all a bit annoying, esp. from a DRY point of view, when it seems as though we could add once and for all,

• ¬[x≢x] : ∀ {w} {Whatever : Set w} →.(x ≢ x) → Whatever in Relation.Binary.PropositionalEquality.Core, encapsulating the use of contradiction-irr in a single place
• (optional) ¬[x≉x] : ∀ {w} {Whatever : Set w} → .(x ≉ x) → Whatever in Relation.Binary.Properties.Setoid by delegation to the above, via reflexive

and hence replace all the existing contradiction refl-like arguments with appeals to ¬[x≢x] instead.

Cons:

• bikeshedding/Fairbairn threshold, trading off against DRY
• seems to go against the spirit, at least, of Prefer contradiction over ⊥-elim when it makes sense? #2653, by re-introducing an implicit appeal to ⊥-elim, esp. wrt the Whatever boilerplate...
• what else?

Pros:

• subtle/slight streamlining of the dependency graph wrt import of Relation.Nullary.Negation.Core
• slight/subtle improvement in the abstraction wrt proofs, esp. being agnostic wrt relevance/irrelevance
• what else?

Suggestions as to names might also be welcome ;-)