oeis_62567_conjecture_0.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 348 curated AI-written annotations and 373 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 12 manuscript-grounded curated result annotations were composed from pinned companion PDFs and named source records in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Twelve supporting declarations for the Hilbert-functions result were upgraded on 2026-10-14 from mechanically condensed doc comments to declaration-specific explanations bound to exact page-one terminology in the pinned human-written companion. The other Hilbert challenge statements were deliberately not described as companion-backed results because that manuscript proves P5 specifically. Fifteen previously unannotated declarations along the Hilbert P5 proof spine were annotated on 2026-10-14 from the pinned companion's seven-page argument, covering the two-generator decomposition, finite-difference bounds, critical-case reduction, and final log-concavity step. Each quotation is bound to its exact manuscript page and Lean declaration; no human review is claimed. Twenty-four further declarations along the Hilbert P5 box-count and finite-difference spine were annotated on 2026-10-14, covering the two-dimensional slice count, three-dimensional box formula, first- and second-difference identities, the seven-threshold curvature expression, boundary handling, pointwise transport, sum linearity, and lower-or-upper region classification. Every annotation is bound to an exact quotation on pages 2 or 3 of the pinned human-written companion; Lean-specific boundary and threshold helpers are identified as implementation lemmas rather than separate manuscript claims. No human review is claimed. Twenty-eight further declarations along the Hilbert P5 argument were annotated on 2026-10-14 from pages 2 through 6 of the pinned human-written companion. They expose the shifted residual sequence, exact lower- and upper-region formulas, symmetry by complementary divisors, the curvature-one and curvature-two splits, and the final algebraic reductions. Lean-only transport and integer-arithmetic helpers are labeled as proof infrastructure rather than new manuscript claims; each quotation is page-bound in the source ledger, and no human review is claimed. Twenty high-level declarations completing the Hilbert P5 curvature-one and curvature-two case spine were annotated on 2026-10-14 from pages 3 through 7 of the pinned human-written companion. The explanations separate cases A and B, lower and upper monomial regions, shifted residual evaluation, impossible mixed regions, and the surviving upper-upper estimate; wrapper lemmas are explicitly described as proof structure. Every quotation is page-bound, and no human review is claimed. Twenty-two graph-reconstruction declarations were upgraded or added on 2026-10-14 from the pinned four-page human-written companion, covering the incidence-deletion deck, edge and degree recovery, unique isolated vertices, vertex-type reconstruction, neighboring-type descent, adjacency recovery, and final bipartite isomorphism. The result annotation also records that `G_not_regular` is present in the Lean statement but unused by its proof term and absent from the companion theorem. Exact historical quotations from Paul J. Kelly (1957) and Bondy with Hemminger (1977) are linked separately; no human review is claimed. Eighteen declarations along the Erdős 741(i) strict-natural-density counterexample were added on 2026-10-14, covering the carry-free base-four digit split, full cross-sum coverage, sparse pure self-sums, fixed-factor fat intervals, the lower-scale density deficit, upper-scale sum growth, and the final convergence contradiction. The result annotation was corrected to distinguish the negative strict-natural-density Lean theorem from the compatible affirmative upper-density companion theorem; Ulam AI's written counterexample is explicitly identified as using a different interval length with the same mechanism. The quotations are page- and hash-bound where the PDFs expose text; Paul Erdős's scanned 1994 question was visually checked and is linked separately without a machine-audit claim. No human review is claimed. Twenty-eight additional declaration-level annotations were added on 2026-10-14 for the Erdős 12(i) proof, and its IsGood definition was upgraded from a source-comment condensation. They cover the prime and Chinese-remainder moduli, the actual Lean congruence-block witness, exclusion of cross-block and same-block triples, lower and upper counting bounds, and the positive liminf. The prior result annotation was corrected because the pinned human-written companion proves the same theorem with a different base-2-to-base-3 construction; it is no longer presented as the Lean proof. The original property-P definition from Erdős and Sárközy (1970) is separately linked and page-bound; no human review is claimed. The remaining fifty declarations in the Erdős 12(i) file were annotated on 2026-10-14, completing declaration-level coverage of all eighty definitions, lemmas, and theorems in that file. These additions expose the prime-recursion invariants, divisibility transport, Chinese-remainder witness bounds, numerical density envelopes, injective block coding, and liminf support lemmas instead of leaving the proof spine as raw Lean; no human review is claimed. Eighteen supporting declarations for Erdős 125 were added on 2026-10-14, completing all nineteen declarations in the file. The annotations follow the pinned companion through digit-block decomposition, irrational logarithmic approximation, the explicit Lean contraction factor 11/12, geometric iteration, and the lower-density conclusion. The original 1996 question of Burr, Erdős, Graham, and Li and the 2026 discussion contributions of GTSoukalas, Terence Tao, and Giuseppe Melfi are linked separately; no human review is claimed. Twenty-eight supporting declarations for Erdős 138 were upgraded or added on 2026-10-14, completing all twenty-nine declarations in the file. The annotations separate the greedy difference proof from the Lean-specific representation bridges and ultrafilter compactness proof of existence, and bind quotations from the pinned companion, the AlphaProof Nexus paper, Erdős's 1981 source, van der Waerden's recollection, and the 2026 Erdős Problems discussion. No human review is claimed. The curated Erdős 12(ii), 138, and 152 annotations were further linked on 2026-10-14 to exact quotations from Felix Behrend, Bartel van der Waerden, and Paul Erdős with Pál Turán; the historical quotation sources are labeled separately from the pinned AlphaProof proof companions. The curated Hilbert-functions result, its OrderIdeal and pureOSequence definitions, and the Erdős 846 annotation were further linked on 2026-10-14 to exact quotations from Richard P. Stanley and from Moe Putterman, Mehtaab Sawhney, and Gregory Valiant. The Stanley PDF is hash- and page-bound in the source ledger; these historical sources are labeled separately from the pinned AlphaProof proof companions. Three additional target-specific annotations for Ben Green's Problem 57, exact last-iterate anchored-GDA convergence, and Graffiti.pc Conjecture 2 were composed on 2026-10-14 from the exact pinned Lean statements, quotations from the AlphaProof Nexus paper, and quotations from Green, Ryu with Yuan and Yin, Surina et al., and Wang with Dai and Tong; no human review is claimed. Two additional target-specific quantum-optics annotations for the explicit AME(11,5) construction and the even N >= 4, D = N monochromatic-quantum-graph theorem were composed on 2026-10-14 from the exact pinned Lean statements, page-bound quotations from the AlphaProof Nexus paper, Helwig et al., and Krenn, Gu, and Zeilinger, and Mario Krenn's public priority correction. The AME annotation makes no first-discovery claim; the monochromatic annotation states that the local AlphaProof wrapper contains sorry and links the separate pinned proof-bearing Formal Conjectures artifact without silently treating it as local verification. Twelve declarations that the pinned source repeats verbatim in several files were annotated in only one of the copies, so the build rejected them as differing copies. On 2026-10-16 each curated annotation was copied unchanged to every identical copy: the eleven Hilbert-functions definitions from hilbert_functions_5 to hilbert_functions_1, _2, _3, _4, _7 and _8, and Erdos12.IsGood from erdos_12.parts.i to erdos_12.parts.ii (the better-sourced annotation was kept in each case). Each copy's quotations are bound in the source ledger like the original's; no new text was written and no human review is claimed. The remaining source-derived annotations were 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 source prose.

Credit: Google DeepMind published the AlphaProof Nexus proof artifacts. The pinned repository identifies the mathematical sources and collaborators, including OEIS, Erdős Problems, Gergely Bérczi, and Mario Krenn; those source records remain the authority for individual credit.

Project sources: Pinned AlphaProof Nexus results documentation, AlphaProof Nexus paper, Pinned human-written natural-language proof companions

35 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

FormalConjectures.Util.ProblemImports

def reverse_nat The number whose digits in base 10 are $n$'s digits reversed line 43 source
Source documentation
The number whose digits in base 10 are $n$'s digits reversed.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.ofDigits, OfNat.ofNat, List.reverse, Nat.digits
2 instances, coercions and auxiliary constants left out.
```
def reverse_nat (k : ℕ) : ℕ :=
…
```
def a A062567: First multiple of $n$ whose reverse is also divisible by $n$, or 0 if no such multiple exists line 49 source
Source documentation
A062567: First multiple of $n$ whose reverse is also divisible by $n$, or 0 if no such multiple exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, And, GT.gt, Dvd.dvd, reverse_nat, HMul.hMul, dite, Exists, DecidablePred, inferInstance, Nat.find, Not
10 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/OEIS/A309132_conjecture_carmichael.lean:45 (same statement text; different proof or body), APNOutputs/OEIS/a091669_conjecture_primitive_root.lean:46 (same statement text; different proof or body), APNOutputs/OEIS/a325046_odd_terms_at_k_times_k_plus_1.lean:52 (different declaration text), APNOutputs/OEIS/oeis_103311_conjecture_0.lean:47 (different declaration text), APNOutputs/OEIS/oeis_108_conjecture_2.lean:45 (different declaration text), APNOutputs/OEIS/oeis_113254_conjecture_0.lean:48 (different declaration text), APNOutputs/OEIS/oeis_175386_conjecture_0.lean:46 (different declaration text), APNOutputs/OEIS/oeis_194806_conjecture_0.lean:50 (same statement text; different proof or body), APNOutputs/OEIS/oeis_227582_conjecture_0.lean:60 (same statement text; different proof or body), APNOutputs/OEIS/oeis_228143_conjecture_1.lean:54 (same statement text; different proof or body), APNOutputs/OEIS/oeis_243106_conjecture_0.lean:47 (different declaration text), APNOutputs/OEIS/oeis_248802_conjecture_0.lean:43 (different declaration text), APNOutputs/OEIS/oeis_248802_conjecture_4.lean:43 (different declaration text), APNOutputs/OEIS/oeis_267581_conjecture_0.lean:71 (same statement text; different proof or body), APNOutputs/OEIS/oeis_271591_conjecture_0.lean:57 (different declaration text), APNOutputs/OEIS/oeis_2897_conjecture_0.lean:46 (different declaration text), APNOutputs/OEIS/oeis_306424_conjecture_0.lean:52 (same statement text; different proof or body), APNOutputs/OEIS/oeis_307865_conjecture_0.lean:46 (different declaration text), APNOutputs/OEIS/oeis_323557_conjecture_0.lean:48 (different declaration text), APNOutputs/OEIS/oeis_341254_conjecture_0.lean:52 (same statement text; different proof or body), APNOutputs/OEIS/oeis_372761_conjecture_2.lean:83 (same statement text; different proof or body), APNOutputs/OEIS/oeis_A028859_conjecture_1.lean:43 (different declaration text), APNOutputs/OEIS/oeis_a211417_conjecture_specific.lean:44 (different declaration text), APNOutputs/OEIS/oeis_a237271_conjecture_2.lean:49 (different declaration text), APNOutputs/OEIS/oeis_a300997_finite_difference_is_one_or_two.lean:46 (same statement text; different proof or body), APNOutputs/OEIS/oeis_a363102_conjecture_1.lean:56 (different declaration text), APNOutputs/OEIS/oeis_a368692_conjecture_integrality.lean:47 (different declaration text). Qualify references as a@path.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
def sum_digits line 100 source
no axioms what this means
Statement uses: List, Nat
Proof uses: sum_digits.match_1, Unit, List.nil, OfNat.ofNat, List.cons, HAdd.hAdd
5 instances, coercions and auxiliary constants left out.
def weight_digits line 104 source
no axioms what this means
Statement uses: List, Nat
Proof uses: sum_digits.match_1, Unit, List.nil, OfNat.ofNat, List.cons, HAdd.hAdd, sum_digits
5 instances, coercions and auxiliary constants left out.
lemma sum_digits_append line 108 source
standard axioms: propext what this means
Statement uses: List, Nat, Eq, sum_digits, HAppend.hAppend, HAdd.hAdd
Proof uses: List.rec, of_eq_true, OfNat.ofNat, Eq.trans, True, congrArg, zero_add, eq_self, congr, add_assoc
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma weight_digits_append line 111 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat, Eq, weight_digits, HAppend.hAppend, HAdd.hAdd, HMul.hMul, List.length, sum_digits
Proof uses: Eq.mpr, id, congrArg, add_assoc, List.rec, sum_digits.match_1, Unit, List.nil, OfNat.ofNat, List.cons, of_eq_true, List.brecOn.go, PUnit, PProd, Eq.trans, True, MulZeroClass.zero_mul, add_zero, zero_add, eq_self and 12 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma sum_digits_reverse line 118 source
standard axioms: propext what this means
Statement uses: List, Nat, Eq, sum_digits, List.reverse
Proof uses: List.rec, Eq.refl, List.nil, List.cons, HAppend.hAppend, List.reverse_cons, HAdd.hAdd, sum_digits_append, rfl, of_eq_true, Eq.trans, True, congr, congrArg, id, Eq.mp, Nat.add_comm, eq_self
4 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma weight_digits_reverse_add line 128 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat, Eq, HAdd.hAdd, weight_digits, List.reverse, HMul.hMul, HSub.hSub, List.length, OfNat.ofNat, sum_digits
Proof uses: List.rec, Eq.refl, List.nil, List.cons, HAppend.hAppend, List.reverse_cons, weight_digits_append, rfl, List.length_reverse, Eq.rec, dite, of_eq_true, Eq.trans, True, congr, congrArg, id, Eq.mp, add_zero, MulZeroClass.zero_mul and 31 more
48 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma ofDigits_mod_81 line 142 source
standard axioms: Quot.sound, propext what this means
Statement uses: List, Nat, Eq, HMod.hMod, Nat.ofDigits, OfNat.ofNat, HAdd.hAdd, sum_digits, HMul.hMul, weight_digits
Proof uses: List.rec, rfl, List.nil, Eq.rec, List.cons, Eq.symm, Eq.trans, Eq.mpr, id, congrArg, weight_digits.eq_2, sum_digits.eq_2, Eq.refl, Nat.modEq_of_dvd, Decidable.byContradiction, Dvd.dvd, Int, HSub.hSub, Not, Nat.ofDigits_cons
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma gcd_81 line 146 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Nat.gcd, HAdd.hAdd, OfNat.ofNat, HMul.hMul
Proof uses: Nat.Coprime.pow_right, Nat.Coprime.symm, Iff.mpr, Nat.Coprime, Not, Dvd.dvd, Nat.Prime.coprime_iff_not_dvd, Nat.prime_three
12 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma coprime_mod_81 line 149 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, HMod.hMod, HMul.hMul, HAdd.hAdd, OfNat.ofNat
Proof uses: Dvd.dvd.modEq_zero_nat, HPow.hPow, Iff.mp, Dvd.dvd, Nat.Coprime.dvd_mul_left, Nat.Coprime.pow_left, Iff.mpr, Nat.Coprime, Not, Nat.Prime.coprime_iff_not_dvd, Nat.prime_three, Nat.dvd_of_mod_eq_zero
19 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma digits_lt_10 Helper lemmas for finite cases line 153 source
Source comment
Helper lemmas for finite cases

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Membership.mem, List, Nat.digits, OfNat.ofNat, LE.le
Proof uses: LT.lt.le_pred, HAdd.hAdd, Nat.digits_lt_base'
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma digits_lt_10 (x : ℕ) (d : ℕ) (h : d ∈ digits 10 x) : d ≤ 9 := by
…
```
lemma sum_digits_bound line 156 source
standard axioms: propext what this means
Statement uses: List, Nat, Membership.mem, LE.le, OfNat.ofNat, sum_digits, HMul.hMul, List.length
Proof uses: List.rec, of_decide_eq_true, List.nil, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, List.cons, LE.le.trans, HAdd.hAdd, Nat.casesAuxOn, of_eq_true, Eq.trans, True, congr, congrArg, add_zero, eq_true and 9 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma length_digits_le line 160 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, HPow.hPow, OfNat.ofNat, LE.le, List.length, Nat.digits
Proof uses: dite, Eq, of_eq_true, Eq.trans, True, List.nil, congrArg, List, Nat.digits_zero, eq_true, True.intro, Eq.symm, Not, Eq.rec, HAdd.hAdd, Nat.log, Nat.log_lt_of_lt_pow, Nat.digits_len, of_decide_eq_true, id and 4 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma ofDigits_of_all_9 line 163 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat, LE.le, List.length, OfNat.ofNat, Membership.mem, GE.ge, sum_digits, Eq, Nat.ofDigits
Proof uses: LT.lt, Or, Or.symm, LE.le.eq_or_lt, Or.inl, False, LE.le.not_gt, HSMul.hSMul, LE.le.trans, sum_digits.match_1, Unit, List.nil, List.cons, HAdd.hAdd, List.sum, ge_of_eq, List.rec, of_decide_eq_true, id, Bool and 35 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma x_lt_10_pow_9 line 172 source
standard axioms: propext what this means
Statement uses: Nat, LT.lt, HSub.hSub, HPow.hPow, OfNat.ofNat
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma a_9 line 174 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, a, OfNat.ofNat
Proof uses: Eq.mpr, dite, Exists, And, GT.gt, Dvd.dvd, reverse_nat, HMul.hMul, Nat.find, inferInstance, DecidablePred, Eq.mpr_prop, Eq.refl, Not, id, congrArg, ite, Eq.trans, ite_cond_eq_false, eq_false' and 42 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma a_27 line 178 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, a, OfNat.ofNat
Proof uses: id, Eq.mpr, ite, And, GT.gt, Dvd.dvd, reverse_nat, HMul.hMul, dite, Exists, DecidablePred, inferInstance, Nat.find, Not, LT.lt, Nat.ofDigits, List.reverse, Nat.digits, Eq.mpr_prop, congrArg and 44 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma no_solution_81 line 185 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, LT.lt, Not, Dvd.dvd, reverse_nat, HMul.hMul
Proof uses: mul_pos, Nat.zero_lt_succ, HSub.hSub, HPow.hPow, Decidable.byContradiction, Eq, HMod.hMod, Nat.mul_mod_left, List, Nat.digits, Nat.ofDigits, Nat.ofDigits_digits, List.reverse, rfl, HAdd.hAdd, sum_digits, weight_digits, ofDigits_mod_81, sum_digits_reverse, List.length and 85 more
71 instances, coercions and auxiliary constants left out.

Tactic trace: 50 steps
Open to load the checked before-and-after goal states.
lemma a_eq_of_min line 217 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, And, Dvd.dvd, reverse_nat, HMul.hMul, LT.lt, Not, Eq, a
Proof uses: Eq.mpr, ite, dite, Exists, Nat.ofDigits, List.reverse, Nat.digits, Nat.find, DecidablePred, Eq.mpr_prop, Eq.refl, id, congrArg, inferInstance, ite_congr, dite_congr, Exists.intro, dif_pos, Iff.mpr, Nat.find_eq_iff and 4 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma a_81_k line 223 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, a, OfNat.ofNat, HMul.hMul
Proof uses: GT.gt, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Dvd.dvd, reverse_nat, Eq.mpr, Nat.ofDigits, List.reverse, Nat.digits, congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_mul, Mathlib.Meta.NormNum.isNat_ofNat, of_eq_true, Eq.trans, HAppend.hAppend and 22 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma a_81 line 234 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, a, OfNat.ofNat
Proof uses: HMul.hMul, a_81_k, Eq.refl, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
def L5 line 240 source
no axioms what this means
Statement uses: List, Nat
Proof uses: List.cons, OfNat.ofNat, List.nil
1 instances, coercions and auxiliary constants left out.
def L_seq line 242 source
no axioms what this means
Statement uses: Nat, List
Proof uses: L_seq.match_1, Unit, OfNat.ofNat, L5, Nat.succ, HAppend.hAppend
5 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean:330 (different declaration text). Qualify references as L_seq@path.
def V line 246 source
standard axioms: propext what this means
Statement uses: Nat
Proof uses: Nat.ofDigits, OfNat.ofNat, L_seq
2 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/OEIS/a325046_odd_terms_at_k_times_k_plus_1.lean:120 (different declaration text). Qualify references as V@path.
lemma L_seq_len line 248 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, List.length, L_seq, HMul.hMul, OfNat.ofNat, HPow.hPow
Proof uses: id, Nat.recAux, List, L_seq.match_1, Unit, L5, Nat.succ, HAppend.hAppend, Eq.refl, Eq.trans, HAdd.hAdd, Eq.mpr, Nat.rec, PProd, PProd.mk, Nat.zero, PUnit, PUnit.unit, congrArg, List.length_append and 31 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma V_step line 252 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, V, HAdd.hAdd, OfNat.ofNat, HMul.hMul, HPow.hPow
Proof uses: Eq.mpr, id, congrArg, add_assoc, Nat.ofDigits, L_seq, V.eq_1, List, L_seq.match_1, Unit, L5, Nat.succ, HAppend.hAppend, List.length, Nat.rec, PProd, PProd.mk, Nat.zero, PUnit, PUnit.unit and 47 more
63 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma three_div_factor line 259 source
standard axioms: propext what this means
Statement uses: Nat, Dvd.dvd, OfNat.ofNat, HAdd.hAdd, HPow.hPow, HMul.hMul
Proof uses: of_eq_true, Eq.trans, Eq, HMod.hMod, True, congrArg, Nat.add_mod, congr, Nat.pow_mod, one_pow, eq_true_of_decide, Eq.refl, Bool, Bool.true
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma V_div line 262 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Dvd.dvd, HPow.hPow, OfNat.ofNat, HAdd.hAdd, V
Proof uses: Eq.mpr, id, Eq, congrArg, Eq.symm, add_comm, Nat.ofDigits, L_seq, V.eq_1, Nat.rec, List, L_seq.match_1, Unit, L5, Nat.succ, HAppend.hAppend, of_decide_eq_true, Nat.zero, Bool, Decidable.decide and 26 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma V_rev line 269 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, reverse_nat, V
Proof uses: id, Nat.ofDigits, OfNat.ofNat, List.reverse, Nat.digits, L_seq, List, Nat.rec, L5, HAppend.hAppend, Trans.simple, L_seq.match_1, Unit, Nat.succ, congr_arg, Function.comp, rfl, Nat.zero, Eq.mpr, congrArg and 53 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
lemma V_pos line 280 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GT.gt, V, OfNat.ofNat
Proof uses: Nat.strongRec, LT.lt, V_pos.match_2, V_pos.match_1, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Star.star, Nat.succ, Nat.add, HAdd.hAdd, Nat.ofDigits, L_seq, Nat.rec, List and 27 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma V_lt line 291 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, V, HSub.hSub, HPow.hPow, OfNat.ofNat, HAdd.hAdd
Proof uses: Eq.mpr, id, Eq, Nat.ofDigits, L_seq, Nat.rec, List, L_seq.match_1, Unit, L5, Nat.succ, HAppend.hAppend, of_decide_eq_true, Nat.zero, Bool, Decidable.decide, Bool.true, Eq.refl, lt_of_le_of_lt, List.length and 29 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma a_le_V line 297 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, a, HPow.hPow, OfNat.ofNat, HAdd.hAdd, V
Proof uses: Eq.mpr, dite, Exists, And, LT.lt, Dvd.dvd, reverse_nat, HMul.hMul, Nat.find, GT.gt, DecidablePred, Eq.mpr_prop, congrArg, funext, congr, add_comm, Not, Nat.ofDigits, L_seq, id and 112 more
112 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.
lemma a_gt_4 Helper lemma for the general inequality when n >= 5 line 318 source
Source comment
Helper lemma for the general inequality when n >= 5

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, LT.lt, a, HPow.hPow, HSub.hSub
Proof uses: Eq, HAdd.hAdd, Decidable.byContradiction, Not, Nat.sub_eq_of_eq_add, V, a_le_V, V_lt, Eq.mpr, eq_of_heq, LE, HEq, Eq.refl, HEq.refl, Eq.symm, outParam, HPow, id, congrArg, LE.le.trans_lt
24 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
lemma a_gt_4 (n : ℕ) (hn : 5 ≤ n) : a (3 ^ n) < 10 ^ 3 ^ (n - 2) - 1 := by
…
```
theorem target_theorem_0 line 329 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Iff, Eq, a, HPow.hPow, HSub.hSub, Or
Proof uses: Iff.intro, LT.lt, lt_trichotomy, Decidable.byContradiction, Not, a_gt_4, False.elim, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, HAdd.hAdd, Neg.neg, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.add_congr, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne, HMul.hMul, Mathlib.Tactic.Ring.instCommSemiringNat, Int.rawCast and 51 more
91 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/OEIS/A224515_conjecture_existence.lean:269 (different declaration text), APNOutputs/OEIS/A309132_conjecture_carmichael.lean:375 (different declaration text), APNOutputs/OEIS/A382590_conjecture_kth_prime_factor_is_eventually_periodic.lean:303 (different declaration text), APNOutputs/OEIS/a091669_conjecture_primitive_root.lean:275 (different declaration text), APNOutputs/OEIS/a325046_odd_terms_at_k_times_k_plus_1.lean:187 (different declaration text), APNOutputs/OEIS/oeis_103311_conjecture_0.lean:240 (different declaration text), APNOutputs/OEIS/oeis_108_conjecture_2.lean:281 (different declaration text), APNOutputs/OEIS/oeis_113254_conjecture_0.lean:153 (different declaration text), APNOutputs/OEIS/oeis_175386_conjecture_0.lean:326 (different declaration text), APNOutputs/OEIS/oeis_194806_conjecture_0.lean:495 (different declaration text), APNOutputs/OEIS/oeis_227582_conjecture_0.lean:306 (different declaration text), APNOutputs/OEIS/oeis_228143_conjecture_1.lean:730 (different declaration text), APNOutputs/OEIS/oeis_243106_conjecture_0.lean:164 (different declaration text), APNOutputs/OEIS/oeis_248802_conjecture_0.lean:223 (different declaration text), APNOutputs/OEIS/oeis_248802_conjecture_4.lean:841 (different declaration text), APNOutputs/OEIS/oeis_256012_conjecture_0.lean:116 (different declaration text), APNOutputs/OEIS/oeis_267581_conjecture_0.lean:212 (different declaration text), APNOutputs/OEIS/oeis_271591_conjecture_0.lean:518 (different declaration text), APNOutputs/OEIS/oeis_278070_conjecture_0.lean:91 (different declaration text), APNOutputs/OEIS/oeis_282779_conjecture_0.lean:141 (different declaration text), APNOutputs/OEIS/oeis_289411_conjecture_0.lean:189 (different declaration text), APNOutputs/OEIS/oeis_2897_conjecture_0.lean:427 (different declaration text), APNOutputs/OEIS/oeis_306424_conjecture_0.lean:300 (different declaration text), APNOutputs/OEIS/oeis_307865_conjecture_0.lean:212 (different declaration text), APNOutputs/OEIS/oeis_323557_conjecture_0.lean:214 (different declaration text), APNOutputs/OEIS/oeis_340737_conjecture_0.lean:456 (different declaration text), APNOutputs/OEIS/oeis_341254_conjecture_0.lean:136 (different declaration text), APNOutputs/OEIS/oeis_363347_conjecture_2.lean:848 (different declaration text), APNOutputs/OEIS/oeis_372761_conjecture_2.lean:756 (different declaration text), APNOutputs/OEIS/oeis_51293_conjecture_0.lean:517 (different declaration text), APNOutputs/OEIS/oeis_A028859_conjecture_1.lean:432 (different declaration text), APNOutputs/OEIS/oeis_A258667_conjecture_0.lean:446 (different declaration text), APNOutputs/OEIS/oeis_a211417_conjecture_specific.lean:264 (different declaration text), APNOutputs/OEIS/oeis_a237271_conjecture_2.lean:124 (different declaration text), APNOutputs/OEIS/oeis_a300997_finite_difference_is_one_or_two.lean:1111 (different declaration text), APNOutputs/OEIS/oeis_a363102_conjecture_1.lean:302 (different declaration text), APNOutputs/OEIS/oeis_a368692_conjecture_integrality.lean:182 (different declaration text). Qualify references as target_theorem_0@path.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.