VCDimConvex.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 8 contain sorry, 0 declared as axioms

VCₙ dimension of convex sets in ℝⁿ, ℝⁿ⁺¹, ℝⁿ⁺²

In the literature it is known that every convex set in ℝ² has VC dimension at most 3, and there exists a convex set in ℝ³ with infinite VC dimension (even more strongly, which shatters an infinite set).

This file states that every convex set in ℝⁿ has finite VCₙ dimension, constructs a convex set in ℝⁿ⁺² with infinite VCₙ dimension (even more strongly, which n-shatters an infinite set), and conjectures that every convex set in ℝⁿ⁺¹ has finite VCₙ dimension.

Imports (1)

FormalConjecturesUtil

lemma VCDimConvex.hasAddVCDimAtMost_three_of_convex_r2 Every convex set in $\mathbb R^2$ has VC dimension at most 3 sorry line 42 source
Source documentation
Every convex set in $\mathbb R^2$ has VC dimension at most 3.

depends on sorry what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Convex, ENNReal, HasAddVCDimAtMost
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.
```
lemma hasAddVCDimAtMost_three_of_convex_r2 {C : Set ℝ²} (hC : Convex ℝ C) : HasAddVCDimAtMost C 3 :=
…
```
lemma VCDimConvex.exists_infinite_convex_r3_shatters There exists a set in $\mathbb R^3$ shattering an infinite set sorry line 47 source
Source documentation
There exists a set in $\mathbb R^3$ shattering an infinite set.

depends on sorry what this means
Statement uses: Exists, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, And, Set.Infinite, Convex, ENNReal, Shatters, Set.ofPred, Eq, HVAdd.hVAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.
```
lemma exists_infinite_convex_r3_shatters :
    ∃ A C : Set ℝ³, A.Infinite ∧ Convex ℝ C ∧ Shatters {t +ᵥ C | t : ℝ³} A := sorry
…
```
Mentions: Shatters
lemma VCDimConvex.exists_convex_rn_add_two_vc_n_forall_not_hasAddVCNDimAtMost There exists a set of infinite $\mathrm{VC}_n$ dimension in $\mathbb R^{n + 2}$ sorry line 54 source
Source documentation
There exists a set of infinite $\mathrm{VC}_n$ dimension in $\mathbb R^{n + 2}$.

depends on sorry what this means
Statement uses: Nat, Exists, Set, Fin, HAdd.hAdd, OfNat.ofNat, Real, And, Convex, Not, HasAddVCNDimAtMost
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.
```
lemma exists_convex_rn_add_two_vc_n_forall_not_hasAddVCNDimAtMost (n : ℕ) :
    ∃ C : Set (Fin (n + 2) → ℝ), Convex ℝ C ∧ ∀ d, ¬ HasAddVCNDimAtMost C n d := sorry
…
```
lemma VCDimConvex.hasAddVCNDimAtMost_two_one_of_convex_r3 sorry line 64 source
depends on sorry what this means
Statement uses: Not, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Convex, ENNReal, HasAddVCNDimAtMost
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.
lemma VCDimConvex.hasAddVCNDimAtMost_two_two_of_convex_r3 Every convex set in $\mathbb R^3$ has $\mathrm{VC}_2$ dimension at most 2 sorry line 70 source
Source documentation
Every convex set in $\mathbb R^3$ has $\mathrm{VC}_2$ dimension at most 2.

depends on sorry what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Convex, ENNReal, HasAddVCNDimAtMost
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.
```
lemma hasAddVCNDimAtMost_two_two_of_convex_r3 {C : Set ℝ³} (hC : Convex ℝ C) :
    HasAddVCNDimAtMost C 2 2 := sorry
…
```
lemma VCDimConvex.exists_hasAddVCNDimAtMost_n_of_convex_rn_add_one sorry line 82 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Set, Fin, HAdd.hAdd, Real, Convex, HasAddVCNDimAtMost
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.
lemma VCDimConvex.exists_exponential_bound_hasAddVCNDimAtMost_n_of_convex_rn_add_one For every $n \ge 1$ there exists some $d$ such that every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most $d$ sorry line 91 source
Source documentation
For every $n \ge 1$ there exists some $d$ such that every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most $d$.
The bound can be taken to be $d = c^n$ for some constant $c \ge 2$ independent of $n$.

depends on sorry what this means
Statement uses: Exists, Nat, And, LE.le, OfNat.ofNat, Set, Fin, HAdd.hAdd, Real, Convex, HasAddVCNDimAtMost, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.
```
lemma exists_exponential_bound_hasAddVCNDimAtMost_n_of_convex_rn_add_one :
    ∃ c : ℕ, 2 ≤ c ∧ ∀ (n : ℕ) (hn : 1 ≤ n),
      ∀ C : Set (Fin (n + 1) → ℝ), Convex ℝ C → HasAddVCNDimAtMost C n (c ^ n) := sorry
…
```
Mentions: c
lemma VCDimConvex.hasAddVCNDimAtMost_n_two_of_convex_rn_add_one Is it true that, for all $n$, every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most 3? sorry line 98 source
Source documentation
Is it true that, for all $n$, every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most 3?

depends on sorry what this means
Statement uses: Iff, True, Nat, Ne, OfNat.ofNat, Set, EuclideanSpace, Real, Fin, HAdd.hAdd, Convex, ENNReal, HasAddVCNDimAtMost
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma hasAddVCNDimAtMost_n_two_of_convex_rn_add_one :
    answer(sorry) ↔ ∀ ⦃n : ℕ⦄, n ≠ 0 → ∀ ⦃C : Set (EuclideanSpace ℝ (Fin (n + 1)))⦄,
      Convex ℝ C → HasAddVCNDimAtMost C n 3 := by
…
```