associativity of ideal product

[Alizter]

This was left as a TODO so I did a quick proof for it. It's possible it can be simplified.

[Alizter]

@jdchristensen I managed to carry out the opposite argument you suggested, but it required a lot of refactoring in Ideal. Namely, exploding the Section IdealLemmas which is probably for the best. While the diff is big, it might be a good time to perform some more style changes like changing Lemma to Definition.

[jdchristensen]

I started reviewing this, but the change to the indentation made it impossible to see what changed later on in the file, as github got quite confused with the diff it displayed. It's better if large indentation changes can be moved to a separate commit. Maybe you can force push to fix this, replacing just the last commit? (I think the comments I already made should still be visible and understandable after this.)

[jdchristensen]

Since this is about Subgroups, not Ideals, it should go in Groups/Subgroup.v. Same with reflexivity, transitivity and antisymmetry. But really, this is just about any predicates on any type, so maybe it just belongs in Types/Forall or Basics/Overture or something like that?

[jdchristensen]

This can be strengthened: If I is an ideal, and J is any predicate on R which is pointwise logically equivalent to I, then J is a subgroup and an ideal.

[Alizter]

Isn't this just the composition of two results then? Any J logically equivalent to a subgroup I is also a subgroup, furthermore also an ideal when I is one.

[jdchristensen]

Yes. But we may as well state the stronger version if we generalize these predicate comparisons to any predicates on any types.

[Alizter]

You're totally right about the diff being hard to read. I've gone ahead and separated that out. Sorry about that. I'll get to your other review suggestions later.

[jdchristensen]

Using opposite rings didn't really save many lines here. (Moreover, with another change I'm going to suggest, the proof of the first case will become a lot shorter, making the use of opposite rings even less efficient here.) This part can be made a bit cleaner using a helper definition:

Definition ideal_product_op_triple {R : Ring} (I J K : Ideal R)
  : ideal_op R ((K ⋅ J) ⋅ I)
      ⊆ (ideal_op R I) ⋅ ((ideal_op R J) ⋅ (ideal_op R K)).
Proof.
  intros x i.
  apply (ideal_product_op (R:=rng_op R)).
  nrapply (ideal_product_subset_pres_l (R:=R)).
  1: nrapply (ideal_product_op (R:=rng_op R)).
  apply i.
Defined.

Then this proof can become:

    apply (ideal_product_op_triple (R:=rng_op R) I J K).
    nrapply f.
    apply ideal_product_op_triple.

(That helper definition is one-sided, though, so it's not a perfect solution. Maybe using transitivity of ideal_eq (or its generalization to predicates on any type) could make it symmetrical?)

[jdchristensen]

I think I'll just push this along with my other idea to see what you think.

[jdchristensen]

I pushed three commits. The first two I think are clear benefits. And subgroup_generated_subset_subgroup can be used in other places in Ideal.v and probably in other places in Subgroup.v or maybe elsewhere. Look for places using ideal_in_zero and ideal_in_plus_negate.

The third commit is the thing I wrote a comment about. Feel free to revert if you think it is not worth it. Or maybe with the new approach to the assertion, it's not worth using opposite rings here?

I do think opposite rings help with ideals in other parts of the file, so that's great. Can you find more places where this helps?

I have to run and won't have time to look at this for a while, but hopefully you'll be able to use some of these ideas to clean things up further.