6.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 6

References:

erdosproblems.com/6
[BFT15] Banks, William D. and Freiberg, Tristan and Turnage-Butterbaugh, Caroline L., Consecutive primes in tuples. Acta Arith. (2015), 261-266.
[Ma15] Maynard, James, Small gaps between primes. Ann. of Math. (2) (2015), 383-413.

Imports (1)

FormalConjecturesUtil

theorem Erdos6.erdos_6 There are infinitely many $n$ such that $d_n < d_{n+1} < d_{n+2}$, where $d$ denotes the prime gap function sorry line 38 source
Source documentation
There are infinitely many $n$ such that $d_n < d_{n+1} < d_{n+2}$, where $d$ denotes the prime gap function.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, And, LT.lt, primeGap, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_6 :
    {n | primeGap n < primeGap (n + 1) ∧ primeGap (n + 1) < primeGap (n + 2)}.Infinite := by
…
```
Mentions: primeGap
theorem Erdos6.erdos_6.variants.increasing For all $m$, there are infinitely many $n$ such that $d_n < d_{n+1} < \dots < d_{n+m}$, where $d$ denotes the prime gap function sorry line 50 source
Source documentation
For all $m$, there are infinitely many $n$ such that $d_n < d_{n+1} < \dots < d_{n+m}$, where $d$ denotes the prime gap function.
Proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]

depends on sorry what this means
Statement uses: Nat, Set.Infinite, Set.ofPred, Membership.mem, Finset, Finset.range, LT.lt, primeGap, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_6.variants.increasing (m : ℕ) :
    {n | ∀ i ∈ Finset.range m, primeGap (n + i) < primeGap (n + i + 1)}.Infinite := by
…
```
Mentions: primeGap
theorem Erdos6.erdos_6.variants.decreasing For all $m$, there are infinitely many $n$ such that $d_n > d_{n+1} \dots > d_{n+m}$, where $d$ denotes the prime gap function sorry line 63 source
Source documentation
For all $m$, there are infinitely many $n$ such that $d_n > d_{n+1} \dots > d_{n+m}$, where $d$ denotes the prime gap function.
Proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]

depends on sorry what this means
Statement uses: Nat, Set.Infinite, Set.ofPred, Membership.mem, Finset, Finset.range, GT.gt, primeGap, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_6.variants.decreasing (m : ℕ) :
    {n | ∀ i ∈ Finset.range m, primeGap (n + i) > primeGap (n + i + 1)}.Infinite := by
…
```
Mentions: primeGap