feat(Order/ConditionallyCompleteLattice): ConditionallyCompleteSemiLatticeInf

[Jun2M]

This PR introduces ConditionallyCompleteSemilatticeSup/Inf, Sup/Inf only versions of ConditionallyCompleteLattice.

This is a preliminary PR for #37620, showing that Graph forms a ConditionallyCompleteSemilatticeInf.

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[mathlib-bors]

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While this PR remains draft, bors will ignore commands on this PR. Mark it ready for review before using commands like bors r+ or bors try.

[github-actions]

PR summary 9036ce5439

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

+ ConditionallyCompleteSemilatticeInf
+ ConditionallyCompleteSemilatticeSup
+ OrderDual.instConditionallyCompleteSemilatticeSup
+ conditionallyCompleteSemilatticeInfOfsInf
+ conditionallyCompleteSemilatticeInfOfsInfTop
+ isGLB_csInf
- isLUB_csSup

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]

The design looks reasonable to me, but do you have any examples of structures that would fit these instances?

[vihdzp]

You could get rid of this by adding a requirement to conditionally complete (semi)lattices that if they have a bottom/top, then the supremum/infimum of the empty set equals to it. I think that'd be a worthwhile refactor which can be done independently of adding these semilattice instances.

[Jun2M]

I am not quite sure how I can say in Lean "if there is bottom/top, so and so is true". Could you spell out exactly what you are thinking?

[vihdzp]

IsBot x -> sSup \empty = x

[vihdzp]

Ah, this is for graphs. You know, surreals under the simplicity ordering also satisfy this typeclass but not ConditionallyCompleteLattice. So I'm happy to have them split up!

[Jun2M]

@vihdzp, Could I ask for some help with making to_dual tag working? I don't really understand what is wrong.

[vihdzp]

I'd like to see this change done in a preliminary PR, which would also deprecate ConditionallyCompleteSemilatticeSupBot.

[vihdzp]

I've opened a Zulip thread to double-check that this is a good idea.

[staroperator]

do you have any examples of structures that would fit these instances?

I encountered another one, the $\alpha^{<\beta}$ tree. (This is probably a sublattice of graphs.)