224515.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

8 declarations, 1 contain sorry, 0 declared as axioms

Existence of integers $k$ with $k^2 \operatorname{XOR} (k+1)^2 = (2n+1)^2$

a: $a(n) = \text{least } k \text{ such that } \sqrt{k^2 \operatorname{XOR} (k+1)^2} = 2n+1$, $a(n) = -1 \text{ if there is no such } k$. This is equivalent to finding the smallest $k \in \mathbb{N}$ such that $k^2 \oplus (k+1)^2 = (2n+1)^2$. We use the set infimum ($\operatorname{sInf}$) to denote the least element of the set of natural numbers satisfying the condition. Since Mathlib's sInf on a subset of ℕ gives a result in ℕ, this definition is only completely faithful to the OEIS when the set is non-empty. The OEIS definition implies that the set of k's is non-empty for all n.

References:

A224515
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA224515.a a: $a(n) = \text{least } k \text{ such that } \sqrt{k^2 \operatorname{XOR} (k+1)^2} = 2n+1$, $a(n) = -1 \text{ if there is no such } k$ line 57 source
Source documentation
a: $a(n) = \text{least } k \text{ such that } \sqrt{k^2 \operatorname{XOR} (k+1)^2} = 2n+1$, $a(n) = -1 \text{ if there is no such } k$. This is equivalent to finding the smallest $k \in \mathbb{N}$ such that $k^2 \oplus (k+1)^2 = (2n+1)^2$. We use the set infimum ($\operatorname{sInf}$) to denote the least element of the set of natural numbers satisfying the condition. Since Mathlib's sInf on a subset of ℕ gives a result in ℕ, this definition is only completely faithful to the OEIS when the set is non-empty. The OEIS definition implies that the set of k's is non-empty for all n, which is legitimate as shown by the theorem exists_xor_sq_eq below.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HPow.hPow, HAdd.hAdd, HMul.hMul, OfNat.ofNat, InfSet.sInf, Set.ofPred, Eq, Nat.xor
10 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
lemma OeisA224515.A224515_eq line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Nat.xor, HPow.hPow, OfNat.ofNat, HAdd.hAdd, HMul.hMul, LT.lt, Ne, OeisA224515.a
Proof uses: id, InfSet.sInf, Set.ofPred, csInf_eq_of_forall_ge_of_forall_gt_exists_lt, Exists.intro, Membership.mem, Set, Decidable.byContradiction, LE.le, Not, And, And.intro
34 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma OeisA224515.a_0 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224515.a, OfNat.ofNat
Proof uses: OeisA224515.A224515_eq, of_decide_eq_true, Nat.xor, HPow.hPow, HAdd.hAdd, HMul.hMul, id, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt
13 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA224515.a_1 line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224515.a, OfNat.ofNat
Proof uses: OeisA224515.A224515_eq, of_decide_eq_true, Nat.xor, HPow.hPow, HAdd.hAdd, HMul.hMul, id, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dite, Ne, LE.le, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast and 4 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA224515.a_2 line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224515.a, OfNat.ofNat
Proof uses: OeisA224515.A224515_eq, of_decide_eq_true, Nat.xor, HPow.hPow, HAdd.hAdd, HMul.hMul, id, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dite, Ne, LE.le, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast and 4 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA224515.a_3 line 92 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224515.a, OfNat.ofNat
Proof uses: OeisA224515.A224515_eq, of_decide_eq_true, Nat.xor, HPow.hPow, HAdd.hAdd, HMul.hMul, id, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dite, Ne, LE.le, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast and 4 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA224515.a_4 line 97 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224515.a, OfNat.ofNat
Proof uses: OeisA224515.A224515_eq, of_decide_eq_true, Nat.xor, HPow.hPow, HAdd.hAdd, HMul.hMul, id, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dite, Ne, LE.le, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast and 4 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA224515.exists_xor_sq_eq sorry line 110 source
depends on sorry what this means
Statement uses: Nat, Exists, Eq, Nat.xor, HPow.hPow, OfNat.ofNat, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.