LeanAide GPT-4 documentation

Module: Mathlib.Combinatorics.SetFamily.Shatter

Finset.mem_shatterer

theorem Finset.mem_shatterer : βˆ€ {Ξ± : Type u_1} [inst : DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±}, s ∈ π’œ.shatterer ↔ π’œ.Shatters s := by sorry

In the context where `Ξ±` is a type with decidable equality, this theorem states that for any set `s` and any set family `π’œ` (both of type `Finset Ξ±`), `s` is in the set of sets shattered by `π’œ` if and only if `π’œ` shatters `s`. Here, a set family `π’œ` is said to shatter a set `s` if for every subset `t` of `s`, there exists a set `u` in `π’œ` such that the intersection of `s` and `u` equals `t`.

More concisely: A set `s` is in the sets shattered by a decidably equal set family `π’œ` if and only if for every subset `t` of `s`, there exists `u` in `π’œ` such that `s ∩ u = t`.

Finset.Shatters.shatterer

theorem Finset.Shatters.shatterer : βˆ€ {Ξ± : Type u_1} [inst : DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±}, π’œ.Shatters s β†’ π’œ.shatterer.Shatters s

This theorem, known as an alias of the reverse direction of `Finset.shatters_shatterer`, asserts that for any type `Ξ±` with decidable equality, any finite set `π’œ` of finite sets of `Ξ±`, and any finite set `s` of `Ξ±`, if `π’œ` shatters `s`, then the shatterer of `π’œ` also shatters `s`. Here, "shattering" is a concept from combinatorics related to the breaking down of a set into its constituent subsets. This theorem establishes a kind of symmetry in the shattering operation.

More concisely: If `Ξ±` has decidable equality, `π’œ` is a finite set of finite sets of `Ξ±`, and `s` is a finite set of `Ξ±` such that `π’œ` shatters `s`, then the shatterer of `π’œ` shatters `s`.

Finset.card_shatterer_le_sum_vcDim

theorem Finset.card_shatterer_le_sum_vcDim : βˆ€ {Ξ± : Type u_1} [inst : DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} [inst_1 : Fintype Ξ±], π’œ.shatterer.card ≀ (Finset.Iic π’œ.vcDim).sum fun k => (Fintype.card Ξ±).choose k

The Sauer-Shelah lemma states that for any type `Ξ±` with decidable equality and for any finite set `π’œ` of finite sets of `Ξ±`, the cardinality of the set of elements that shatter `π’œ` is less than or equal to the sum over all `k` from `0` to the Vapnik-Chervonenkis (VC) dimension of `π’œ` of the binomial coefficient "choose `k`" of the cardinality of `Ξ±`. In other words, the number of different ways we can partition `π’œ` using a binary classifier is bounded by the sum of the binomial coefficients up to the VC dimension of `π’œ`.

More concisely: The cardinality of the set of shatterings of a finite set of elements with decidable equality in a Vapnik-Chervonenkis class of dimension d is at most the sum of the binomial coefficients from 0 to d.

Finset.card_le_card_shatterer

theorem Finset.card_le_card_shatterer : βˆ€ {Ξ± : Type u_1} [inst : DecidableEq Ξ±] (π’œ : Finset (Finset Ξ±)), π’œ.card ≀ π’œ.shatterer.card

This is a statement of Pajor's variant of the Sauer–Shelah lemma. For any type `Ξ±` with decidable equality, given a finite set `π’œ` of subsets of `Ξ±`, the size of `π’œ` is less than or equal to the size of its 'shatterer'. Here, the 'shatterer' of `π’œ` is a set that 'shatters' `π’œ`, meaning it intersects with every subset in `π’œ` in a uniquely identifiable way.

More concisely: For any finite collection `π’œ` of subsets of a type `Ξ±` with decidable equality, the size of `π’œ` is bounded by the size of its shatter function, which is the cardinality of the largest set intersecting each subset in `π’œ` in a unique point.

Finset.vcDim_compress_le

theorem Finset.vcDim_compress_le : βˆ€ {Ξ± : Type u_1} [inst : DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)), (Down.compression a π’œ).vcDim ≀ π’œ.vcDim

The theorem `Finset.vcDim_compress_le` states that for any type `Ξ±` with decidable equality, and any element `a` of type `Ξ±` and a finite set `π’œ` of finite sets of `Ξ±`, the Vapnik-Chervonenkis (VC) dimension of the set obtained by down-compressing `π’œ` with respect to `a` is always less than or equal to the VC dimension of `π’œ` itself. In other words, the operation of down-compressing a set does not increase its VC dimension. The down-compression of a set `π’œ` with respect to an element `a` refers to the operation of removing `a` from all members of `π’œ` where it exists, and adding the resulting set to `π’œ` if it is not already present. The VC dimension, a concept from computational learning theory, measures the complexity of a set of sets in terms of its ability to shatter other sets.

More concisely: For any type `Ξ±` with decidable equality and any finite set `π’œ` of finite sets of `Ξ±`, the VC dimension of the down-compressed set of `π’œ` with respect to an element `a` is less than or equal to the VC dimension of `π’œ`.

Mathlib.Combinatorics.SetFamily.Shatter._auxLemma.5

theorem Mathlib.Combinatorics.SetFamily.Shatter._auxLemma.5 : βˆ€ {Ξ± : Type u_1} [inst : DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±}, (s ∈ π’œ.shatterer) = π’œ.Shatters s

The theorem `Mathlib.Combinatorics.SetFamily.Shatter._auxLemma.5` states that for any type `Ξ±` with decidable equality, any set family `π’œ` (a `Finset` of `Finset Ξ±`), and any set `s` (a `Finset Ξ±`), `s` is part of the set family that is shattered by `π’œ` (i.e., `s` is in `Finset.shatterer π’œ`) if and only if `π’œ` shatters `s` (denoted as `Finset.Shatters π’œ s`). Here, `π’œ` shatters `s` means that for every subset `t` of `s`, there exists an element `u` in `π’œ` such that the intersection of `s` and `u` equals `t`. This theorem essentially links the concept of a set being in the shattered set family of `π’œ` with the property of `π’œ` shattering `s`.

More concisely: A set `s` is in the shattered set family of a set family `π’œ` if and only if `π’œ` shatters `s`, i.e., for every subset `t` of `s`, there exists an element `u` in `π’œ` such that `s ∩ u = t`.