feat(QuantumMechanics): add algebraic properties for mulOperator

[gloges]

Adds some basic results about multiplication operators and smul/neg/add/sub:

• c • 𝓜 μ f ≤ 𝓜 μ (c • f) with equality when c ≠ 0,
• 𝓜 μ (-f) = -𝓜 μ f,
• 𝓜 μ f + 𝓜 μ g ≤ 𝓜 μ (f + g) (and a sub variant).

[github-actions]

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[nateabr]

This looks good to me. One comment is that it would be good to have more documentation than is available at the moment. Some doc strings for key lemmas, and a general elaboration on "why" behind the code would be great for readability and would improve the file greatly.

[gloges]

Thanks, I'll aim to improve the documentation for this and other QM files in the near future.

[nateabr]

Okayy sounds good, just one more question:

For smul and composition, the PR (and existing code) follow a nice proof pattern. However, for add, we only get the inequality 𝓜 f + 𝓜 g ≤ 𝓜 (f + g). Would it be worth adding something like mulOperator_add_eq for the case where both domains are ⊤? Maybe a todo?

[nateabr]

awaiting-author

[gloges]

That's a good point. The condition c ≠ 0 is particularly simple for mulOperator_smul_eq but for add/sub it seems much messier. I've added mulOperator_add_eq (and a sub variant) which uses the hypothesis h : (𝓜 μ g).domain = ⊤ and a TODO to expand on this; I'm not sure at this point what conditions would be most useful to pin down.

[nateabr]

Okayy looks good to me, will tag as reviewer approved :)