VCDim.lean`

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L4DEL: Lean4 Domain Explanation Language

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16 declarations, 0 contain sorry, 0 declared as axioms

VC dimension of a set family

This file defines the VC dimension of a set family as the maximum size of a set it shatters.

Imports (1)

Mathlib.Data.Set.Card

Imported by 2 files

def Shatters A set family 𝒜 shatters a set A if restricting 𝒜 to A gives rise to all subsets of A line 33 source
Source documentation
A set family 𝒜 shatters a set A if restricting 𝒜 to A gives rise to all subsets of A.

no axioms what this means
Statement uses: Set
Proof uses: LE.le, Exists, And, Membership.mem, Eq, Inter.inter
3 instances, coercions and auxiliary constants left out.
```
def Shatters : Prop := ∀ ⦃B⦄, B ⊆ A → ∃ C ∈ 𝒜, A ∩ C = B
```
lemma Shatters.exists_inter_eq_singleton line 35 source
no axioms what this means
Statement uses: Set, Shatters, Membership.mem, Exists, And, Eq, Inter.inter, Singleton.singleton
Proof uses: Iff.mpr, LE.le, Set.singleton_subset_iff
6 instances, coercions and auxiliary constants left out.
lemma Shatters.mono_left line 38 source
no axioms what this means
Statement uses: Set, LE.le, Shatters
Proof uses: Exists, And, Membership.mem, Eq, Inter.inter, Exists.intro, And.intro
5 instances, coercions and auxiliary constants left out.
lemma Shatters.mono_right line 41 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, LE.le, Shatters
Proof uses: And, Membership.mem, Eq, Inter.inter, Exists, LE.le.trans, Exists.intro, And.intro, inf_congr_right, inf_le_of_left_le
16 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Shatters.exists_superset line 44 source
standard axioms: Quot.sound, propext what this means
Statement uses: Set, Shatters, Exists, And, Membership.mem, LE.le
Proof uses: Eq, Inter.inter, Set.Subset.rfl, Exists.intro, And.intro, Iff.mp, Set.inter_eq_left
6 instances, coercions and auxiliary constants left out.
lemma Shatters.of_forall_subset line 47 source
standard axioms: Quot.sound, propext what this means
Statement uses: Set, LE.le, Membership.mem, Shatters
Proof uses: Exists.intro, And, Eq, Inter.inter, And.intro, Iff.mpr, Set.inter_eq_right
5 instances, coercions and auxiliary constants left out.
lemma Shatters.nonempty line 50 source
no axioms what this means
Statement uses: Set, Shatters, Set.Nonempty
Proof uses: Exists, And, Membership.mem, Eq, Inter.inter, Set.Subset.rfl, Exists.intro
3 instances, coercions and auxiliary constants left out.
lemma shatters_empty line 53 source
standard axioms: Quot.sound, propext what this means
Statement uses: Set, Iff, Shatters, EmptyCollection.emptyCollection, Set.Nonempty
Proof uses: Iff.intro, Shatters.nonempty, LE.le, Exists, And, Membership.mem, Eq, Inter.inter, Exists.intro, And.intro, Eq.mpr, id, Eq.trans, congrFun', congrArg, Set.empty_inter, Eq.mp
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Shatters.subset_iff line 56 source
no axioms what this means
Statement uses: Set, Shatters, Iff, LE.le, Exists, And, Membership.mem, Eq, Inter.inter
Proof uses: Iff.intro, Set.inter_subset_left
9 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma shatters_iff line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Iff, Shatters, Eq, Set.image, Inter.inter, Set.powerset
Proof uses: Iff.intro, Set.ext, of_eq_true, Membership.mem, Eq.trans, Exists, And, True, congr, congrArg, LE.le, propext, Shatters.subset_iff, iff_self, Eq.superset
13 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma univ_shatters line 62 source
standard axioms: Quot.sound, propext what this means
Statement uses: Set, Shatters, Set.univ
Proof uses: Shatters.of_forall_subset, of_eq_true, LE.le, Membership.mem, Eq.trans, True, forall_congr, implies_congr, Eq.refl, implies_true
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma shatters_univ line 64 source
standard axioms: Quot.sound, propext what this means
Statement uses: Set, Iff, Shatters, Set.univ, Eq
Proof uses: of_eq_true, Eq.trans, Membership.mem, True, congr, congrArg, forall_congr, LE.le, Exists, And, Inter.inter, implies_congr, funext, congrFun', Set.univ_inter, iff_self
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def HasVCDimAtMost Documented declaration: HasVCDimAtMost line 71 source
Source documentation
A set family 𝒜 has VC dimension at most d if there are no families x of elements indexed by [d + 1] and A of sets of 𝒜 indexed by 2^[d + 1] such that x i ∈ A s ↔ i ∈ s for all i ∈ [d + 1], s ⊆ [d + 1].

no axioms what this means
Statement uses: Set, Nat
Proof uses: Fin, HAdd.hAdd, OfNat.ofNat, Membership.mem, Not, Iff
4 instances, coercions and auxiliary constants left out.
```
def HasVCDimAtMost : Prop :=
…
```
lemma HasVCDimAtMost.anti line 75 source
no axioms what this means
Statement uses: Set, Nat, LE.le, HasVCDimAtMost
Proof uses: Fin, HAdd.hAdd, OfNat.ofNat, Membership.mem
6 instances, coercions and auxiliary constants left out.
lemma HasVCDimAtMost.mono line 78 source
standard axioms: Quot.sound, propext what this means
Statement uses: Set, Nat, LE.le, HasVCDimAtMost
Proof uses: Fin, HAdd.hAdd, OfNat.ofNat, Membership.mem, Iff, Decidable.byContradiction, Not, Function.comp, Fin.castLE, Set.image, of_eq_true, Eq.trans, True, forall_congr, eq_true, implies_true, Iff.trans, congrFun', congrArg, Exists and 4 more
11 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma HasVCDimAtMost.empty line 84 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, HasVCDimAtMost, EmptyCollection.emptyCollection, Set
Proof uses: of_eq_true, Fin, HAdd.hAdd, OfNat.ofNat, Membership.mem, Not, Iff, Eq.trans, True, forall_congr, False, Exists, implies_congr, congrArg, funext, implies_true
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.