feat(LinearAlgebra/Basis/HasCanonicalBasis): propose HasCanonicalBasis class

[Paul-Lez]

Context: in some cases, e.g. writing down PDEs, having a notion of partial derivatives can be useful. In an ideal world (as mentioned by @ocfnash in this Zulip thread), one could write something along the lines of open scoped PartialDerivatives in order to access nice notation for this.

This PR introduces a notion of a canonical basis for a module in order to allow taking partial derivatives with respect to it.
A later PR (which I am about open) would then propose a notation for taking partial derivatives (defined using lineDeriv in order to avoid requiring Frechet-differentiability assumptions). The goal here is to have notation that works equally well on e.g. 𝕜 × 𝕜 and EuclideanSpace 𝕜 (Fin 2) and so on.

I've opened this PR as a draft in order to get some initial feedback on the content (and whether this is considered appropriate for Mathlib!); the file itself would need to be split (e.g. the instances defined should probably go in different files)

Co-authored-by: Eric Wieser efw@google.com

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[github-actions]

PR summary c6a8ffddfe

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.LinearAlgebra.Basis.HasCanonicalBasis (new file)	1975

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Declarations diff

+ EuclideanSpace.hasCanonicalBasis
+ HasCanonicalBasis
+ Pi.hasCanonicalBasis
+ basis_apply
+ instance : HasCanonicalBasis 𝕜 (𝕜 × 𝕜 × 𝕜) (Fin 3)
+ instance : HasCanonicalBasis 𝕜 (𝕜 × 𝕜) (Fin 2) (![(1, 0), (0, 1)])
+ instance : HasCanonicalBasis 𝕜 𝕜 (Fin 1) (fun _ ↦ 1)
+ instance [Ring 𝕜] (p : ENNReal) :
+ prod
+ reindex

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[leanprover-community-bot-assistant]

This pull request has conflicts, please merge master and resolve them.

[Paul-Lez]

This PR has been migrated to a fork-based workflow: #39253