[Merged by Bors] - feat(LinearAlgebra/Simplex): lemma for subset/disjoint relation between interior of simplex and its faces

[wwylele]

Intermediate lemma towards #34826. This will be used to show the measure on interior and closedInterior are equal, as the difference is a zero-measure set.

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[github-actions]

PR summary 9b51fe6ec0

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

+ closedInterior_diff_interior
+ closedInterior_eq_interior_union
+ closedInterior_faceOpposite_ssubset_closedInterior
+ closedInterior_faceOpposite_subset_closedInterior
+ closedInterior_face_ssubset_closedInterior
+ closedInterior_face_subset_closedInterior
+ disjoint_interior_closedInterior_face
+ disjoint_interior_closedInterior_faceOpposite
+ point_mem_closedInterior_faceOpposite_iff
+ point_mem_closedInterior_face_iff

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).

[Copilot]

Pull request overview

Adds an intermediate lemma in Affine.Simplex relating closedInterior to interior plus the union of the closedInteriors of codimension-1 faces (faceOpposite), intended for later measure-theoretic arguments in #34826.

Changes:

• Prove (s.face h).closedInterior ⊆ s.closedInterior.
• Prove s.closedInterior = s.interior ∪ ⋃ i, (s.faceOpposite i).closedInterior under ordered-ring assumptions.

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[wwylele]

@eric-wieser Does this look good now?

[jcommelin]

Thanks 🎉

bors merge

[mathlib-bors]

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[mathlib-bors]

This PR was included in a batch that was canceled, it will be automatically retried

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

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