[Merged by Bors] - feat(Order): WithBotTop and the extended integers

[joelriou]

This files defines an abbreviation WithBotTop ι for WithBot (WithTop ι).
We also introduce an abbreviation EInt for WithBotTop ℤ.

Kevin Buzzard is made a coauthor of the file because it is very similar to Data.EReal.Basic.

---

relevant Zulip topic

[github-actions]

PR summary ec57dd4792

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Order.WithBotTop (new file)	171

---

Declarations diff

+ EInt
+ WithBotTop
+ WithBotTop.coe
+ coe_injective
+ coe_le_coe
+ coe_lt_coe
+ coe_monotone
+ coe_ne_bot
+ coe_ne_top
+ coe_strictMono
+ instance : Coe ι (WithBotTop ι) := ⟨WithBotTop.coe⟩
+ rec
+ rec_bot
+ rec_coe
+ rec_top
+ top_ne_bot

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).

[vihdzp]

You don't have to do it here, but would there be any value in defining Extended X or WithBotTop X as WithBot (WithTop X) so as to standardize a single spelling for this concept?

[joelriou]

You don't have to do it here, but would there be any value in defining Extended X or WithBotTop X as WithBot (WithTop X) so as to standardize a single spelling for this concept?

I have generalized the lemmas to a more general setting, and made EInt a particular case of that.

[joneugster]

From the Zulip discussion, I gather that the dual version of WithBotTop gives a different value for ⊥ + ⊤ and that there have been refactors previously exactly to make the value of this behave better.

It seems that this choice is driven by actual use cases in the library. I suggest to leave the question about how - if at all - WithTop (WithBot ι) should be integrated into the existing API to future work. And having WithBotTop merged might even help in that development.

Thank you for the PR! I suggest

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by joneugster.

[jcommelin]

Thanks 🎉

bors merge

[mathlib-bors]

Pull request successfully merged into master.

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