[Merged by Bors] - chore(topology/metric_space/pi_Lp): move to analysis folder, import inner_product_space #7991
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…nner_product_space (#7991) Currently, the file `pi_Lp` (on finite products of metric spaces, with the `L^p` norm) is in the topology folder, but it imports a lot of analysis (to have real powers) and it defines a normed space structure, so it makes more sense to have it in analysis. Also, it is currently imported by `inner_product_space`, to give an explicit construction of an inner product space on `pi_Lp 2`, which means that all files importing general purposes lemmas on inner product spaces also import real powers, trigonometry, and so on. We swap the imports, letting `pi_Lp` import `inner_product_space` and moving the relevant bits from the latter file to the former. This gives a more reasonable import graph.
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Currently, the file
pi_Lp(on finite products of metric spaces, with theL^pnorm) is in the topology folder, but it imports a lot of analysis (to have real powers) and it defines a normed space structure, so it makes more sense to have it in analysis. Also, it is currently imported byinner_product_space, to give an explicit construction of an inner product space onpi_Lp 2, which means that all files importing general purposes lemmas on inner product spaces also import real powers, trigonometry, and so on. We swap the imports, lettingpi_Lpimportinner_product_spaceand moving the relevant bits from the latter file to the former. This gives a more reasonable import graph.No math changes at all. Strangely, a proof (which was alrady very slow before the PR) has become just a bit slower after removing an import and times out, so I had to squeeze its simps. Possibly the removed import has a new simp lemma that is efficient in this proof.