[ add ] injectivity of suc for relation m ≡ n (mod o) for {{NonZero o}}

[jamesmckinna]

Taken from the discussion at https://agda.zulipchat.com/#narrow/channel/264623-stdlib/topic/suc.20injective.20under.20_.25_/with/582024092 :

• introduces definition m ≲%[ o ] n = ∃ λ k → n ≡ m + k * o
• characterises the relation m % o ≡ n % o in terms of (the symmetric closure of) m ≲%[ o ] n
• gives two proofs of 'Carette's Lemma' (suc m) % o ≡ (suc n) % o → m % o ≡ n % o
  • indirect, via the characterisation, using a new lemma about SymClosure
  • direct, via a slick proof due to @alexarice

Possible TODO:

• prove that _≲%[ o ]_ satisfies IsPreorder _≡_, indeed IsPartialOrder _≡_ etc. (decidable, ...)
• relate this to Modular arithmetic in terms of ideals #2729 and friends, esp. Create a version of [m+kn]%n≡m%n for integers #2877

[jamesmckinna]

Last commit tidies up notations etc.
Still can't quite make the proof carettesLemma delegate via a suitable combinator to ≲%[o]⇒≡[o]% ∘ ≲%[o]-suc⁻¹ but this will have to do, I guess. Suggestions welcome!

[jamesmckinna]

Last commit tidies up notations etc. Still can't quite make the proof carettesLemma delegate via a suitable combinator to ≲%[o]⇒≡[o]% ∘ ≲%[o]-suc⁻¹ but this will have to do, I guess. Suggestions welcome!

Can now: by generalising SymClosure.gmap! Final commit closes this last niggle.

[jamesmckinna]

This proof could be refactored in terms of Relation.Binary.Consequences.wlog, but my first attempt ended up more verbose and harder to read than this version. But perhaps eventually that combinator will be reimplemented, and with it, a suitable refactoring of this lemma.

[jamesmckinna]

Notwithstanding that wlog can be used, I think the above proof is easier to read.

[jamesmckinna]

And now, in terms of Sum elimination, so avoiding the appeal to with.

[jamesmckinna]

@JacquesCarette hope I've addressed all your feedback
@gallais feel free to insist on the wlog proof, but I think the direct one is better!