feat: CountableSupClosed

[RemyDegenne]

Define the property for a set of being closed by countable supremum (resp. infimum).
The new file is adapted from the SupClosed file, which describes sets closed by binary supremum.

Also use to_dual on SupClosed.

CountableInfClosed will be used in measure theory, for developments related to compact systems used for Kolmogorov's extension theorem and Choquet's capacitability theorem.

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[github-actions]

PR summary 42a89fc2b9

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Order.CountableSupClosed (new file)	540

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Declarations diff

+ CountableInfClosed
+ CountableInfClosed.dual
+ CountableSupClosed
+ CountableSupClosed.countableSupClosure_eq
+ CountableSupClosed.directedOn
+ CountableSupClosed.dual
+ CountableSupClosed.finsetSup'_mem
+ CountableSupClosed.finsetSup_mem
+ CountableSupClosed.iInter
+ CountableSupClosed.iSup_mem
+ CountableSupClosed.inter
+ CountableSupClosed.prod
+ CountableSupClosed.sInter
+ CountableSupClosed.sSup_mem
+ CountableSupClosed.supClosed
+ CountableSupClosed.univ
+ InfClosed.countableSupClosure
+ SupClosed.countableInfClosure
+ countableInfClosed_preimage_ofDual
+ countableInfClosed_preimage_toDual
+ countableInfClosure
+ countableSupClosed_countableSupClosure
+ countableSupClosed_preimage_ofDual
+ countableSupClosed_preimage_toDual
+ countableSupClosed_singleton_bot
+ countableSupClosure
+ countableSupClosure_eq_self
+ countableSupClosure_idem
+ countableSupClosure_min
+ countableSupClosure_mono
+ countableSupClosure_prod
+ countableSupClosure_singleton_bot
+ countableSupClosure_univ
+ finsetSup'_mem_countableSupClosure
+ iSup_mem_countableSupClosure
+ isLUB_countableSupClosure
+ mem_countableSupClosure_iff
+ subset_countableSupClosure
+ supClosed_countableSupClosure
+ sup_mem_countableSupClosure
+ upperBounds_countableSupClosure
- InfClosed
- InfClosed.finsetInf'_mem
- InfClosed.finsetInf_mem
- InfClosed.image
- InfClosed.insert_lowerBounds
- InfClosed.insert_upperBounds
- InfClosed.inter
- InfClosed.preimage
- InfClosed.prod
- IsLowerSet.infClosed
- infClosed_empty
- infClosed_iInter
- infClosed_pi
- infClosed_range
- infClosed_sInter
- infClosed_singleton
- infClosed_univ

You can run this locally as follows

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

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

The doc-module for scripts/pr_summary/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/reporting/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]

Are we only interested in the complete lattice case? Otherwise, I think it'd be best to write this down using IsLUB.

[RemyDegenne]

I'm only interested in the case of sets of sets, so personally complete lattice is enough. If you have another more general definition in mind that does not require too much additional work I'd be happy to apply a suggestion.

[vihdzp]

My idea is to use something like ∀ t ⊆ s, t.Nonempty → t.Countable → ∃ x ∈ s, IsLUB t x.

[vihdzp]

Should be easier to define this as the intersection of all CountableSupClosed sets containing s.