Brennanconjecture.lean`

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

11 declarations, 4 contain sorry, 0 declared as axioms

Brennan's Conjecture

Reference:

Imports (1)

FormalConjecturesUtil

def BrennanConjecture.unitDisk line 35 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Complex
Proof uses: Set.ofPred, LT.lt, Real, Norm.norm, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
structure BrennanConjecture.IsUnivalentNormalized The standard class $\mathcal{S}$ of normalised univalent functions on $\mathbb{D}$ line 38 source
Source documentation
The standard class $\mathcal{S}$ of normalised univalent functions on $\mathbb{D}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex
```
structure IsUnivalentNormalized (f : ℂ → ℂ) : Prop where
…
```
def BrennanConjecture.integralMeansSpectrum $\beta_f(\tau) := \limsup_{r \to 1^-} \frac{\log \int_{-\pi}^{\pi} |f'(re^{i\theta})|^\tau \, d\theta}{|\log(1-r)|}$ line 46 source
Source documentation
$\beta_f(\tau) := \limsup_{r \to 1^-} \frac{\log \int_{-\pi}^{\pi} |f'(re^{i\theta})|^\tau \, d\theta}{|\log(1-r)|}$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Real
Proof uses: Filter.limsup, HDiv.hDiv, Real.log, MeasureTheory.integral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.Ioc, Neg.neg, Real.pi, HPow.hPow, Norm.norm, deriv, HSMul.hSMul, Complex.exp, HMul.hMul, Complex.I, abs, HSub.hSub, OfNat.ofNat, nhdsWithin and 1 more
52 instances, coercions and auxiliary constants left out.
```
noncomputable def integralMeansSpectrum (f : ℂ → ℂ) (τ : ℝ) : ℝ :=
…
```
def BrennanConjecture.universalSpectrum line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: SupSet.sSup, Set.ofPred, Exists, Complex, And, BrennanConjecture.IsUnivalentNormalized, Eq, BrennanConjecture.integralMeansSpectrum
1 instances, coercions and auxiliary constants left out.
def BrennanConjecture.universalSpectrumBounded line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: SupSet.sSup, Set.ofPred, Exists, Complex, And, BrennanConjecture.IsUnivalentNormalized, Bornology.IsBounded, Set.image, BrennanConjecture.unitDisk, Eq, BrennanConjecture.integralMeansSpectrum
7 instances, coercions and auxiliary constants left out.
theorem BrennanConjecture.universalSpectrumBounded_le sorry line 61 source
depends on sorry what this means
Statement uses: Real, LE.le, BrennanConjecture.universalSpectrumBounded, BrennanConjecture.universalSpectrum
Proof uses: csSup_le_csSup, Set.ofPred, Exists, Complex, And, BrennanConjecture.IsUnivalentNormalized, Bornology.IsBounded, Set.image, BrennanConjecture.unitDisk, Eq, BrennanConjecture.integralMeansSpectrum, sorryAx, Lean.Name, BddAbove, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat and 7 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem BrennanConjecture.integralMeansSpectrum_id sorry line 69 source
depends on sorry what this means
Statement uses: Real, Eq, BrennanConjecture.integralMeansSpectrum, id, Complex, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem BrennanConjecture.brennan_universalSpectrum Brennan's conjecture, part 1: $B(-2) = 1$ sorry line 74 source
Source documentation
Brennan's conjecture, part 1: $B(-2) = 1$.

depends on sorry what this means
Statement uses: Eq, Real, BrennanConjecture.universalSpectrum, Neg.neg, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem brennan_universalSpectrum :
    universalSpectrum (-2) = 1 := by
…
```
Mentions: BrennanConjecture.universalSpectrum
theorem BrennanConjecture.brennan_universalSpectrumBounded Brennan's conjecture, part 2: $B_b(-2) = 1$ sorry line 80 source
Source documentation
Brennan's conjecture, part 2: $B_b(-2) = 1$.

depends on sorry what this means
Statement uses: Eq, Real, BrennanConjecture.universalSpectrumBounded, Neg.neg, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem brennan_universalSpectrumBounded :
    universalSpectrumBounded (-2) = 1 := by
…
```
Mentions: BrennanConjecture.universalSpectrumBounded
theorem BrennanConjecture.brennan_spectra_eq Brennan's conjecture, part 3: $B(-2) = B_b(-2)$ line 86 source
Source documentation
Brennan's conjecture, part 3: $B(-2) = B_b(-2)$.

depends on sorry what this means
Statement uses: Eq, Real, BrennanConjecture.universalSpectrum, Neg.neg, OfNat.ofNat, Nat, BrennanConjecture.universalSpectrumBounded
Proof uses: Eq.mpr, id, congrArg, BrennanConjecture.brennan_universalSpectrum, BrennanConjecture.brennan_universalSpectrumBounded, Eq.refl
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem brennan_spectra_eq :
    universalSpectrum (-2) = universalSpectrumBounded (-2) := by
…
```
Mentions: BrennanConjecture.universalSpectrum, BrennanConjecture.universalSpectrumBounded
theorem BrennanConjecture.brennan Brennan's conjecture: $B(-2) = B_b(-2) = 1$ line 92 source
Source documentation
Brennan's conjecture: $B(-2) = B_b(-2) = 1$.

depends on sorry what this means
Statement uses: And, Eq, Real, BrennanConjecture.universalSpectrum, Neg.neg, OfNat.ofNat, Nat, BrennanConjecture.universalSpectrumBounded
Proof uses: And.intro, BrennanConjecture.brennan_universalSpectrum, BrennanConjecture.brennan_universalSpectrumBounded
16 instances, coercions and auxiliary constants left out.
```
theorem brennan :
    universalSpectrum (-2) = 1 ∧ universalSpectrumBounded (-2) = 1 :=
…
```
Mentions: BrennanConjecture.universalSpectrum, BrennanConjecture.universalSpectrumBounded