Documentation

Mathlib.Combinatorics.SetFamily.Shatter

Shattering families #

This file defines the shattering property and VC-dimension of set families.

Main declarations #

TODO #

source
def Finset.Shatters {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) (s : Finset α) :
Prop

A set family 𝒜 shatters a set s if all subsets of s can be obtained as the intersection of s and some element of the set family, and we denote this 𝒜.Shatters s. We also say that s is traced by 𝒜.

Equations
    Instances For
      source
      instance Finset.instDecidablePredShatters {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} :
      DecidablePred 𝒜.Shatters
      Equations
        source
        theorem Finset.Shatters.exists_inter_eq_singleton {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} (hs : 𝒜.Shatters s) (ha : a ∈ s) :
        ∃ t ∈ 𝒜, s ∩ t = {a}
        source
        theorem Finset.Shatters.mono_left {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} (h : 𝒜 ⊆ ℬ) (h𝒜 : 𝒜.Shatters s) :
        ℬ.Shatters s
        source
        theorem Finset.Shatters.mono_right {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s t : Finset α} (h : t ⊆ s) (hs : 𝒜.Shatters s) :
        𝒜.Shatters t
        source
        theorem Finset.Shatters.exists_superset {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (h : 𝒜.Shatters s) :
        ∃ t ∈ 𝒜, s ⊆ t
        source
        theorem Finset.shatters_of_forall_subset {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (h : ∀ t ⊆ s, t ∈ 𝒜) :
        𝒜.Shatters s
        source
        theorem Finset.Shatters.nonempty {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (h : 𝒜.Shatters s) :
        𝒜.Nonempty
        source
        @[simp]
        theorem Finset.shatters_empty {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} :
        𝒜.Shatters ∅ ↔ 𝒜.Nonempty
        source
        theorem Finset.Shatters.subset_iff {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s t : Finset α} (h : 𝒜.Shatters s) :
        t ⊆ s ↔ ∃ u ∈ 𝒜, s ∩ u = t
        source
        theorem Finset.shatters_iff {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} :
        𝒜.Shatters s ↔ image (fun (t : Finset α) => s ∩ t) 𝒜 = s.powerset
        source
        theorem Finset.univ_shatters {α : Type u_1} [DecidableEq α] {s : Finset α} [Fintype α] :
        univ.Shatters s
        source
        @[simp]
        theorem Finset.shatters_univ {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} [Fintype α] :
        𝒜.Shatters univ ↔ 𝒜 = univ
        source
        def Finset.shatterer {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) :
        Finset (Finset α)

        The set family of sets that are shattered by 𝒜.

        Equations
          Instances For
            source
            @[simp]
            theorem Finset.mem_shatterer {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} :
            s ∈ 𝒜.shatterer ↔ 𝒜.Shatters s
            source
            theorem Finset.shatterer_mono {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} (h : 𝒜 ⊆ ℬ) :
            𝒜.shatterer ⊆ ℬ.shatterer
            source
            theorem Finset.subset_shatterer {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} (h : IsLowerSet ↑𝒜) :
            𝒜 ⊆ 𝒜.shatterer
            source
            @[simp]
            theorem Finset.isLowerSet_shatterer {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) :
            IsLowerSet ↑𝒜.shatterer
            source
            @[simp]
            theorem Finset.shatterer_eq {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} :
            𝒜.shatterer = 𝒜 ↔ IsLowerSet ↑𝒜
            source
            @[simp]
            theorem Finset.shatterer_idem {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} :
            𝒜.shatterer.shatterer = 𝒜.shatterer
            source
            @[simp]
            theorem Finset.shatters_shatterer {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} :
            𝒜.shatterer.Shatters s ↔ 𝒜.Shatters s
            source
            theorem Finset.Shatters.shatterer {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} :
            𝒜.Shatters s → 𝒜.shatterer.Shatters s

            Alias of the reverse direction of Finset.shatters_shatterer.

            source
            theorem Finset.card_le_card_shatterer {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) :
            𝒜.card ≤ 𝒜.shatterer.card

            Pajor's variant of the Sauer-Shelah lemma.

            source
            theorem Finset.Shatters.of_compression {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} (hs : (Down.compression a 𝒜).Shatters s) :
            𝒜.Shatters s
            source
            theorem Finset.shatterer_compress_subset_shatterer {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) :
            (Down.compression a 𝒜).shatterer ⊆ 𝒜.shatterer

            Vapnik-Chervonenkis dimension #

            source
            def Finset.vcDim {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) :
            ℕ

            The Vapnik-Chervonenkis dimension of a set family is the maximal size of a set it shatters.

            Equations
              Instances For
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                theorem Finset.vcDim_mono {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} (h𝒜ℬ : 𝒜 ⊆ ℬ) :
                𝒜.vcDim ≤ ℬ.vcDim
                source
                theorem Finset.Shatters.card_le_vcDim {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (hs : 𝒜.Shatters s) :
                s.card ≤ 𝒜.vcDim
                source
                theorem Finset.vcDim_compress_le {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) :
                (Down.compression a 𝒜).vcDim ≤ 𝒜.vcDim

                Down-compressing decreases the VC-dimension.

                source
                theorem Finset.card_shatterer_le_sum_vcDim {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} [Fintype α] :
                𝒜.shatterer.card ≤ ∑ k ∈ Iic 𝒜.vcDim, (Fintype.card α).choose k

                The Sauer-Shelah lemma.