feat(Combinatorics/SetFamily): Assouad's dual VC bound

[Zetetic-Dhruv]

Adds the Finset-level form of Assouad's 1983 dual VC bound: if a family
𝒜 : Finset (Finset α) has VC dimension at most d, then for any
ground set X : Finset α the dual family
{𝒜.filter (· ∋ x) : x ∈ X} has VC dimension at most 2 ^ (d + 1) - 1.

New declarations (in Finset namespace):

• dualFamily 𝒜 X: for each x ∈ X, the subfamily {A ∈ 𝒜 | x ∈ A}
• mem_dualFamily (@[simp]): membership characterisation
• exists_shatters_of_dualFamily_shatters: Assouad's bitstring-coding lemma
• vcDim_dualFamily_le: the headline VC bound

Proof by Assouad's classical bitstring-coding argument. Sits on top of Finset.shatterer / Finset.vcDim from
Mathlib.Combinatorics.SetFamily.Shatter.

References:

• P. Assouad, Densite et dimension, Ann. Inst. Fourier 33(3) (1983), Thm 2.13
• J. Matousek, Lectures on Discrete Geometry, GTM 212, Springer, 2002, Section 10.3 Lemma 10.3.3

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[github-actions]

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[github-actions]

PR summary 98261a9c88

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Combinatorics.SetFamily.DualVC (new file)	580

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Declarations diff

+ dualFamily
+ exists_shatters_of_dualFamily_shatters
+ mem_dualFamily
+ vcDim_dualFamily_le

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).

[Zetetic-Dhruv]

@Shreyas4991 I have added the PR. Can you review it?

[Shreyas4991]

Will do sometime in the next 24 hours.

[Zetetic-Dhruv]

Okay, thanks a lot. Let me know if I need to update anything.