1223.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

7 declarations, 1 contain sorry, 0 declared as axioms

Prime gaps

Differences between consecutive primes: $a(n) = p_{n+1} - p_n$.

References:

A001223

Imports (1)

FormalConjecturesUtil

def OeisA1223.a Prime gaps: differences between consecutive primes line 34 source
Source documentation
Prime gaps: differences between consecutive primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HSub.hSub, Nat.nth, Nat.Prime
4 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
def OeisA1223.gapSubsequence Helper definition for extracting a finite subsequence (pattern) as a list line 39 source
Source documentation
Helper definition for extracting a finite subsequence (pattern) as a list.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, List
Proof uses: List.map, OeisA1223.a, HAdd.hAdd, List.range
2 instances, coercions and auxiliary constants left out.
```
noncomputable def gapSubsequence (startIndex length : ℕ) : List ℕ :=
…
```
theorem OeisA1223.a_0 Value of the sequence a at 0 line 44 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1223.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by rfl
…
```
Mentions: OeisA1223.a
theorem OeisA1223.a_1 Value of the sequence a at 1 line 48 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1223.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HSub.hSub, Nat.nth, Nat.Prime, congrArg, Nat.nth_prime_one_eq_three, Nat.nth_prime_zero_eq_two, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA1223.a
theorem OeisA1223.a_2 Value of the sequence a at 2 line 54 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1223.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HSub.hSub, Nat.nth, Nat.Prime, congrArg, Nat.nth_prime_two_eq_five, Nat.nth_prime_one_eq_three, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 2 := by
…
```
Mentions: OeisA1223.a
theorem OeisA1223.a_3 Value of the sequence a at 3 line 60 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1223.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HSub.hSub, Nat.nth, Nat.Prime, congrArg, Nat.nth_prime_three_eq_seven, Nat.nth_prime_two_eq_five, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 2 := by
…
```
Mentions: OeisA1223.a
theorem OeisA1223.conjecture sorry line 73 source
depends on sorry what this means
Statement uses: Not, Nat, GE.ge, OfNat.ofNat, Set.Infinite, Set.ofPred, Eq, List, OeisA1223.gapSubsequence, HAdd.hAdd
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.