Lacunary.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

6 declarations, 0 contain sorry, 0 declared as axioms

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

def IsLacunary Say a sequence is lacunary if there exists some $\lambda > 1$ such that $a_{n+1}/a_n > \lambda$ for all sufficiently large $n$ line 29 source
Source documentation
Say a sequence is lacunary if there exists some $\lambda > 1$ such that $a_{n+1}/a_n > \lambda$ for all sufficiently large $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, LT.lt, HMul.hMul, HAdd.hAdd, Filter.atTop
11 instances, coercions and auxiliary constants left out.
```
def IsLacunary (n : ℕ → ℕ) : Prop := ∃ c > (1 : ℝ), ∀ᶠ k in atTop, c * n k < n (k + 1)
```
Mentions: c
lemma IsLacunary.eventually_lt Every lacunary sequence is eventually strictly increasing line 32 source
Source documentation
Every lacunary sequence is eventually strictly increasing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, IsLacunary, Filter.Eventually, LT.lt, HAdd.hAdd, OfNat.ofNat, Filter.atTop
Proof uses: Real, And, GT.gt, HMul.hMul, LE.le, Exists, Iff.mp, Filter.eventually_atTop, Iff.mpr, Exists.intro, Eq.mp, Eq.trans, congrFun', congrArg, one_mul, forall_imp, Mathlib.Tactic.GCongr.imp_mono, lt_imp_lt_of_le_of_le, mul_le_mul_of_nonneg_right, le_of_lt and 2 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma IsLacunary.eventually_lt {n : ℕ → ℕ} (hn : IsLacunary n) : ∀ᶠ k in atTop, n k < n (k + 1) := by
…
```
Mentions: IsLacunary
def IsLacunaryReal The real-valued analogue of IsLacunary line 40 source
Source documentation
The real-valued analogue of IsLacunary.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Exists, And, GT.gt, OfNat.ofNat, Filter.Eventually, LT.lt, HMul.hMul, HAdd.hAdd, Filter.atTop
9 instances, coercions and auxiliary constants left out.
```
def IsLacunaryReal (a : ℕ → ℝ) : Prop :=
…
```
lemma isLacunary_iff_isLacunaryReal An ℕ-valued sequence is lacunary iff its real cast is lacunary as a real sequence line 44 source
Source documentation
An ℕ-valued sequence is lacunary iff its real cast is lacunary as a real sequence.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, IsLacunary, IsLacunaryReal, Real
Proof uses: Iff.rfl
2 instances, coercions and auxiliary constants left out.
```
lemma isLacunary_iff_isLacunaryReal {n : ℕ → ℕ} :
    IsLacunary n ↔ IsLacunaryReal (fun k => (n k : ℝ)) := Iff.rfl
```
Mentions: IsLacunary, IsLacunaryReal
lemma IsLacunaryReal.eventually_lt A positive lacunary real-valued sequence is eventually strictly increasing line 48 source
Source documentation
A positive lacunary real-valued sequence is eventually strictly increasing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, IsLacunaryReal, Filter.Eventually, LT.lt, OfNat.ofNat, Filter.atTop, HAdd.hAdd
Proof uses: And, GT.gt, HMul.hMul, Filter.mp_mem, Set.ofPred, Membership.mem, Set, Filter.univ_mem', id, lt_of_not_ge, LE.le, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, HPow.hPow, Nat.rawCast, Int.rawCast and 51 more
83 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma IsLacunaryReal.eventually_lt {a : ℕ → ℝ} (ha : IsLacunaryReal a)
    (hpos : ∀ᶠ k in atTop, 0 < a k) : ∀ᶠ k in atTop, a k < a (k + 1) := by
…
```
Mentions: IsLacunaryReal
def IsSublacunary line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Filter.Tendsto, Real, HDiv.hDiv, HAdd.hAdd, OfNat.ofNat, Filter.atTop, nhds
14 instances, coercions and auxiliary constants left out.