Modular arithmetic in terms of ideals

[Taneb]

Opening this PR to share my WIP. I've got a messy proof of the Chinese remainder theorem for arbitrary rings, but in porting it from my standalone library to this I've somehow made some parameters not infer properly

[JacquesCarette]

The new Kernel file looks nice.

[JacquesCarette]

Would you want some help to get this further along?

[Taneb]

Yes, actually. I've been working on a module for the special case of ideals of the ring of integers, and I've been struggling to prove that (for a non-zero modulus) it's finite, which I think it important for the "yes this is modular arithmetic as you know it" feel. I'll post a WIP commit shortly

[JacquesCarette]

Ok, once my students make further progress on the ones they are currently working on, I'll get them to look at this.

[jamesmckinna]

And, ... having murdered my darlings #2257 #2292 I should turn my attention to helping with this!

[jamesmckinna]

My first thought on reading this code is that .{{_ : NonZero m}} should in fact also be part of the module parametrisation, and thus

• the anonymous module _ .{{_ : NonZero m}} where can be removed
• _≋?_ just is _≋?′_

[jamesmckinna]

My second thought is that, for general Rings:

• arithmetic mod 0 does make sense; quotienting by the zero ideal R/(0) is isomorphic to R?
• arithmetic mod 1 also makes sense; R/(1) is the trivial Ring

In either case, for sure the equality on the quotient would be decidable, but perhaps requires that we analyse these three cases separately:

• m = 0
• m = 1
• NonTrivial m (and that might involve defining that notion for ℤ on the model of NonZero?).

So the question is really: do you/we want to consider these special cases for ℤ/mℤ, or would it be 'better' to qualify the module with .{{_ : NonTrivial m}} and infer {{NonZero m}} where needed?

[Taneb]

I'm not sure why we need to treat the m = 1 case specially. It's not a particularly interesting case, but it should work the same as any other nonzero modulus

[jamesmckinna]

Fair enough, but the path to

_≋?_ with ℕ.nonZero? ∣ m ∣
... | yes p = _≋?′_ {{p}}
... | no ¬p = {!!}

is then via _≟_ : DecidableEquality ℤ having first shown R/(0) iso to R for any R : Ring _ _?
(And that ¬p implies m ≡ 0...)

[Taneb]

Yes, that was my plan

[jamesmckinna]

Some errors thrown up by checking with agda-v2.8.0 (beyond the failing tests above):

• Data.Fin.Properties on this branch still has the now-erroneous --warn=noUserWarning which should be fixed after a rebase/merge with the current master?
• Data.Integer.Properties now triggers a warning about a null rewrite step on L1417 (which I hadn't seen caught anywhere else? but the offending line doesn't appear any more on master, so...)