[Merged by Bors] - feat: local inverse of a diffeomorphism

[grunweg]

Add IsLocalDiffeomorphAt.localInverse, a (choice of) local inverse for a local diffeomorphism at x. Prove basic API about it: if f is a local diffeomorphism at x, we prove that

• f and its local inverse near x are local inverses (in both directions),
• composing f and its local inverse is EventuallyEq to the identity near x.
• smoothness results for the local inverse.

PR #8738 will use this to prove that the differential of a local diffeomorphism is a continuous linear equivalence.

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I wondered if adding the analogous IsLocalDiffeomorph.localInverseAt would be useful: if so, I'll be happy to add it.

[github-actions]

PR summary d0e79dbd13

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ PartialDiffeomorph.isLocalDiffeomorphAt
+ contmdiffOn_localInverse
+ localInverse
+ localInverse_contMDiffAt
+ localInverse_contMDiffOn
+ localInverse_eqOn_left
+ localInverse_eqOn_right
+ localInverse_eventuallyEq_left
+ localInverse_eventuallyEq_right
+ localInverse_isLocalDiffeomorphAt
+ localInverse_left_inv
+ localInverse_mdifferentiableAt
+ localInverse_mem_source
+ localInverse_mem_target
+ localInverse_open_source
+ localInverse_right_inv

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).

[jcommelin]

Thanks 🎉

bors merge

[mathlib-bors]

Pull request successfully merged into master.

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