feat: extended charts are local diffeomorphisms on their source #9273
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…rentials: - make variables implicit; - make more variables implicit and re-use some other variables.
Move Diffeomorph.toPartialDiffeomorph further down.
Touch up the docs; introduce PartialDiffeomorph.toLocalHomeomorph.
and PartialHomeomorph.continuous_{to,inv}Fun to continuousOn_{to,inv}Fun.
might need to read up on the variable command.
This is mathematically cleaner (I show injectivity and surjectivity of the differential separately), cleaner in Lean (it's good practice to avoid heavy tactic blocks in definitions) and makes all proofs by rfl work. This requires completeness of the tangent spaces; this is slightly unfortunate, as it was automatic in the previous approaches. Opinions welcome. I'm a bit unsure about the local instances; we'll see if this can be avoided.
Unsure if the extChartAt statements are really necessary...
These are only one rewrite away and don't feel worth it, for me.
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The analogous result for un-extended charts is shown in #8160.
Unresolved questions
are all the tiny API lemmas useful; are any more missing?
do I want similar results for
extChartAt? (In any case, they are easy to deduce.)depends on: feat: differential of a local diffeomorphism is a continuous linear equivalence #8738 for simplicity