[Merged by Bors] - feat: initial segment commutes with Iio

[vihdzp]

---

[github-actions]

PR summary 507929a61a

Import changes exceeding 2%

%	File
+8.38%	Mathlib.Order.InitialSeg

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Order.InitialSeg	346	375	+29 (+8.38%)

Import changes for all files

Files	Import difference
Mathlib.Order.InitialSeg	29

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Declarations diff

++ image_Iio

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.

---

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).

[eric-wieser]

bors d=@YaelDillies

Just in case I'm missing something here

[mathlib-bors]

✌️ YaelDillies can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

[YaelDillies]

bors merge

Thanks!

[YaelDillies]

Sorry, I just noticed this increases imports to quite a lot of files. Can you rework the import tree?

bors r-

[mathlib-bors]

Canceled.

[vihdzp]

It increases them by only 6 imports, to be fair. This seems like a reasonable home for these theorems to me, since OrderEmbedding.image_Iio was already basically saying the same thing. I can't see an obvious alternative besides placing these two theorems in their own file, which seems awkward.

[YaelDillies]

Upper and lower sets are supposed to be very basic objects that are used widely (as you can tell by the import diff, they really are used widely). I don't think initial and principal segments enjoy the same status. So, if anything, initial and principal segments should import upper and lower sets, rather than the other way around.

[vihdzp]

Didn't I already tried doing that previously? I recall that the import difference of doing that was much larger. Let me try...

[YaelDillies]

Sorry if I dropped the ball here 😓, but I don't recall

[vihdzp]

Yeah, this same issue came up when I tried adding IsLowerSet (range f) to this file. My fix back then was to move the definition of IsLowerSet to a more basic file, but that fix won't work here.

That being said, it doesn't seem like this larger import increase actually impacts any other files. So between adding 2 more imports to 600 or so files, or adding 29 more imports to a single file, I think I know which option I prefer.

[YaelDillies]

it doesn't seem like this larger import increase actually impacts any other files

That corroborates my theory that principal/initial segments are nichely used. If you want to get rid of that import increase later on, the next step would be to reduce imports to Order.UpperLower.Basic which would be a useful endeavour regardless.

bors merge

[mathlib-bors]

Pull request successfully merged into master.

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