oeis_372761_conjecture_2.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

47 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

FormalConjectures.Util.ProblemImports

def continued_fraction_tail Recursive function to compute $A_k(n)$, the denominator tail $k - \frac{k+1}{A_{k+1}(n)}$ line 46 source
Source documentation
Recursive function to compute $A_k(n)$, the denominator tail $k - \frac{k+1}{A_{k+1}(n)}$. The base case is at $k = n - 1$, where $A_{n-1} = (n-1) - \frac{n}{n+4}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: WellFounded.Nat.fix, HSub.hSub, InvImage, LT.lt, dite, GE.ge, OfNat.ofNat, Eq, HDiv.hDiv, HAdd.hAdd, Not, And, LE.le
20 instances, coercions and auxiliary constants left out.
```
noncomputable def continued_fraction_tail (n : ℕ) : ℕ → ℚ
| k =>
  if n ≥ 4 then
    if k = n - 1 then
      (n - 1 : ℚ) - (n : ℚ) / (n + 4 : ℚ)
    else if 3 ≤ k ∧ k < n - 1 then
      let k_succ_val := continued_fraction_tail n (k + 1)
…
```
def continued_fraction_val The total value of the continued fraction $C_n$ line 65 source
Source documentation
The total value of the continued fraction $C_n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: ite, LE.le, OfNat.ofNat, Eq, HSub.hSub, HDiv.hDiv, continued_fraction_tail
10 instances, coercions and auxiliary constants left out.
```
noncomputable def continued_fraction_val (n : ℕ) : ℚ :=
…
```
def a A372761: Denominator of the continued fraction $$ \frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{n+4}}}}}} $$ line 83 source
Source documentation
A372761: Denominator of the continued fraction $$ \frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{n+4}}}}}} $$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, LT.lt, OfNat.ofNat, Rat.den, continued_fraction_val
3 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_62567_conjecture_0.lean:49 (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.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
def sum_fact line 122 source
no axioms what this means
Statement uses: Nat
Proof uses: sum_fact.match_1, Unit, OfNat.ofNat, Nat.succ, HAdd.hAdd, Nat.factorial
5 instances, coercions and auxiliary constants left out.
def IntQ2 line 126 source
no axioms what this means
Statement uses: Nat, Int
Proof uses: HAdd.hAdd, HMul.hMul, HSub.hSub, OfNat.ofNat, sum_fact, Nat.factorial
11 instances, coercions and auxiliary constants left out.
def Q_explicit line 129 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: ite, Eq, OfNat.ofNat, HSub.hSub, HMul.hMul, HDiv.hDiv, IntQ2, sum_fact, Nat.factorial
14 instances, coercions and auxiliary constants left out.
lemma Q_explicit_recurrence line 135 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, Rat, Q_explicit, HSub.hSub, HMul.hMul, HAdd.hAdd
Proof uses: id, Q_explicit_recurrence.match_1, ite, HDiv.hDiv, IntQ2, sum_fact, Nat.factorial, of_eq_true, Eq.trans, True, False, congr, congrArg, ite_congr, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_natSub, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool and 105 more
150 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma tail_eq_base line 143 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, Rat, continued_fraction_tail, HSub.hSub, HDiv.hDiv, Q_explicit
Proof uses: id, Eq.rec, dite, HAdd.hAdd, Not, And, LE.le, LT.lt, InvImage, WellFounded.Nat.fix, ite, HMul.hMul, IntQ2, sum_fact, Nat.factorial, tail_eq_base.match_1, Function.comp, Eq.trans, dif_pos, rfl and 96 more
138 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma tail_eq_desc_0 line 150 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, Rat, continued_fraction_tail, HSub.hSub, HDiv.hDiv, Q_explicit
Proof uses: id, Eq.rec, dite, HAdd.hAdd, Not, And, LE.le, LT.lt, InvImage, WellFounded.Nat.fix, ite, HMul.hMul, IntQ2, sum_fact, Nat.factorial, tail_eq_base.match_1, Function.comp, Eq.trans, dif_pos, rfl and 96 more
138 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Q_explicit_pos line 157 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, LE.le, HSub.hSub, GT.gt, Rat, Q_explicit
Proof uses: id, Eq.mpr, ite, Eq, HMul.hMul, HDiv.hDiv, IntQ2, sum_fact, Nat.factorial, LT.lt, HAdd.hAdd, Eq.trans, False, congrArg, ite_congr, Not, eq_false, mt, LE.le.trans_eq, Mathlib.Meta.NormNum.isNat_le_false and 40 more
138 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma Q_explicit_nz line 165 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, LE.le, HSub.hSub, Ne, Rat, Q_explicit
Proof uses: id, ite, Eq, HMul.hMul, HDiv.hDiv, IntQ2, sum_fact, Nat.factorial, Eq.mpr, Not, HAdd.hAdd, Int, sum_fact.match_1, Unit, Nat.succ, And, Nat.brecOn.go, Eq.trans, Or, congrArg and 65 more
153 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma tail_unfold line 173 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, LE.le, LT.lt, HSub.hSub, Ne, Rat, continued_fraction_tail, HAdd.hAdd, Eq, HDiv.hDiv
Proof uses: id, Eq.rec, dite, Not, And, InvImage, WellFounded.Nat.fix, Function.comp, Eq.trans, dif_pos, dif_neg, LT.lt.ne, And.intro, Eq.symm, WellFounded.Nat.fix_eq
35 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma tail_eq_desc_step_aux line 179 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, LE.le, LT.lt, HSub.hSub, Eq, Rat, continued_fraction_tail, HAdd.hAdd, HDiv.hDiv, Q_explicit, Ne
Proof uses: Eq.mpr, HMul.hMul, ite, IntQ2, sum_fact, Nat.factorial, id, congrArg, Eq.trans, div_div_eq_mul_div, sub_div', Not, congr_arg, tail_eq_desc_step_aux.match_1, congr, False, Nat.cast_add, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat and 94 more
130 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma tail_eq_desc line 188 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, LE.le, HSub.hSub, Eq, Rat, continued_fraction_tail, HDiv.hDiv, Q_explicit
Proof uses: Nat.recAux, Decidable.byContradiction, Not, Eq.mpr, id, congrArg, tail_eq_desc_0, HAdd.hAdd, Eq.mp, rfl, LT.lt, Ne, Q_explicit_nz, Or.resolve_right, Iff.mp, Or, div_eq_zero_iff, cast, Eq.trans, Eq.symm and 3 more
57 instances, coercions and auxiliary constants left out.

Tactic trace: 40 steps
Open to load the checked before-and-after goal states.
lemma tail_eq line 220 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, LE.le, HSub.hSub, Eq, Rat, continued_fraction_tail, HDiv.hDiv, Q_explicit
Proof uses: Decidable.byContradiction, Not, tail_eq_desc, Eq.mp, congrArg
17 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma val_eq_unfold line 229 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Ne, Rat, continued_fraction_tail, HSub.hSub, HDiv.hDiv, Eq, continued_fraction_val
Proof uses: id, val_eq_unfold.match_1, Star.star, Decidable.rec, LE.le, Decidable, Not, ite, HAdd.hAdd, if_neg
32 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Q2_eq line 235 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Rat, Q_explicit, OfNat.ofNat, HDiv.hDiv, IntQ2
Proof uses: of_eq_true, ite, HSub.hSub, HMul.hMul, sum_fact, Nat.factorial, HAdd.hAdd, Int, Eq.trans, True, False, congr, congrArg, ite_congr, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false and 22 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma val_eq_aux line 238 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, Rat, continued_fraction_tail, HDiv.hDiv, Q_explicit, Ne, HSub.hSub, HMul.hMul
Proof uses: Eq.mpr, id, congrArg, div_div_eq_mul_div, sub_div', one_div_div, Eq.refl
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma val_eq line 245 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, Rat, continued_fraction_val, HDiv.hDiv, Q_explicit
Proof uses: continued_fraction_tail, tail_eq, Decidable.byContradiction, LE.le, Not, HSub.hSub, Ne, Q_explicit_nz, HMul.hMul, HAdd.hAdd, Q_explicit_recurrence, rfl, Eq.mp, congrArg, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add and 13 more
53 instances, coercions and auxiliary constants left out.

Tactic trace: 38 steps
Open to load the checked before-and-after goal states.
lemma a_val_aux line 274 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, Rat, continued_fraction_val, HDiv.hDiv, Q_explicit, HSub.hSub, HMul.hMul, IntQ2, Int
Proof uses: of_eq_true, Eq.trans, True, congr, congrArg, div_div, Int.cast_mul, Int.cast_ofNat, Int.cast_sub, Int.cast_natCast, eq_self
66 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma a_val line 281 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, a, Rat.den, HDiv.hDiv, Rat, IntQ2, HMul.hMul, Int, HSub.hSub
Proof uses: continued_fraction_val, Q_explicit, val_eq, rfl, Q2_eq, a_val_aux, Eq.mpr, ite, LT.lt, id, congrArg, a.eq_1, Not, of_eq_true, Eq.trans, True, ite_cond_eq_false, eq_false, eq_self, Eq.refl
33 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/OEIS/oeis_227582_conjecture_0.lean:101 (different declaration text), APNOutputs/OEIS/oeis_341254_conjecture_0.lean:100 (different declaration text). Qualify references as a_val@path.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma rat_den_divides line 293 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Ne, OfNat.ofNat, Dvd.dvd, Nat, Rat.den, HDiv.hDiv, Rat, Int.natAbs
Proof uses: cast, Rat.divInt, Eq.symm, congrArg, Iff.mp, Int.natCast_dvd, Rat.den_dvd
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma a_eq_p_implies_p_divides line 296 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Nat.Prime, Ne, Eq, a, Dvd.dvd, Int, HSub.hSub, HMul.hMul
Proof uses: LT.lt, Or, eq_or_lt_of_le, Eq.mpr, Rat.den, continued_fraction_val, id, congrArg, Eq.mp, Eq.trans, ite, ite_cond_eq_false, of_eq_true, Rat, LE.le, HDiv.hDiv, continued_fraction_tail, True, False, ite_congr and 46 more
93 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma intQ2_mod_p line 310 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, HSub.hSub, HMul.hMul, Exists, Int, IntQ2, HAdd.hAdd, Nat.factorial
Proof uses: Eq.mpr, Dvd.dvd, sum_fact, id, Eq.trans, congrArg, funext, dvd_of_mul_left_dvd, Exists.intro, Eq.mp, congr, Nat.cast_sub, of_eq_true, LE.le, eq_true, LE.le.trans', mul_right_mono, eq_true_of_decide, Eq.refl, Bool and 3 more
61 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma p_dvd_intQ2_of_ge line 315 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, HAdd.hAdd, OfNat.ofNat, Nat.Prime, Dvd.dvd, Int, HSub.hSub, HMul.hMul, IntQ2
Proof uses: id, Dvd.dvd.add, sum_fact, Nat.factorial, Dvd.dvd.mul_right, Dvd.dvd.mul_left, cast, Eq.symm, Iff.mpr, LE.le, Nat.Prime.dvd_factorial, Nat.le_sub_of_add_le
35 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma p_sq_dvd line 320 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Nat.Prime, Eq, a, Dvd.dvd, Int, IntQ2, HPow.hPow, HMul.hMul, HSub.hSub
Proof uses: Rat.den, HDiv.hDiv, Rat, a_val, GT.gt, Decidable.byContradiction, Not, Ne, cast, Eq.trans, congrArg, Eq.symm, Nat.cast_zero, Int.cast_zero, Ne.symm, LT.lt.ne, Eq.mp, rfl, Eq.mpr, id and 54 more
112 instances, coercions and auxiliary constants left out.

Tactic trace: 61 steps
Open to load the checked before-and-after goal states.
lemma p_ge_7 line 378 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Ne, OfNat.ofNat, GE.ge
Proof uses: p_ge_7.match_1, absurd, of_decide_eq_false, Eq.refl, Bool, Decidable.decide, HAdd.hAdd, Decidable.byContradiction, Not
9 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma cross_mul_eq line 381 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, a, Exists, Int, And, HMul.hMul, HSub.hSub, IntQ2, Nat.Coprime, Int.natAbs
Proof uses: Rat.den, HDiv.hDiv, Rat, a_val, GT.gt, Decidable.byContradiction, Not, Ne, cast, Eq.trans, congrArg, Eq.symm, Nat.cast_zero, Int.cast_zero, Ne.symm, LT.lt.ne, Eq.mp, rfl, Eq.mpr, id and 56 more
102 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.
lemma p_sq_dvd_5n_4 line 415 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Nat.Prime, Ne, Eq, a, HAdd.hAdd, Dvd.dvd, Int, HPow.hPow, HSub.hSub, HMul.hMul
Proof uses: a_eq_p_implies_p_divides, Decidable.byContradiction, Not, IntQ2, p_dvd_intQ2_of_ge, Exists, And, Nat.Coprime, Int.natAbs, cross_mul_eq, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Mathlib.Tactic.Ring.instCommSemiringNat, Mathlib.Tactic.Ring.pow_congr, Mathlib.Tactic.Ring.atom_pf, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.pow_add, Mathlib.Tactic.Ring.single_pow and 40 more
65 instances, coercions and auxiliary constants left out.

Tactic trace: 42 steps
Open to load the checked before-and-after goal states.
lemma p_pow_dvd_intQ2 line 456 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Nat.Prime, Dvd.dvd, Int, HPow.hPow, HSub.hSub, HMul.hMul, IntQ2
Proof uses: Eq.mpr, HAdd.hAdd, sum_fact, Nat.factorial, id, Eq, congrArg, IntQ2.eq_1, Dvd.dvd.add, Dvd.dvd.mul_right, mul_comm, Eq.mp, Eq.symm, Dvd.dvd.mul_left, dvd_trans, Nat.casesAuxOn, absurd, of_decide_eq_false, Eq.refl, Bool and 162 more
187 instances, coercions and auxiliary constants left out.

Tactic trace: 52 steps
Open to load the checked before-and-after goal states.
lemma p_pow_dvd_5n_4_step line 482 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Nat.Prime, Ne, Eq, a, Dvd.dvd, Int, HPow.hPow, HSub.hSub, HMul.hMul, HAdd.hAdd
Proof uses: p_ge_7, IntQ2, p_pow_dvd_intQ2, Exists, And, Nat.Coprime, Int.natAbs, cross_mul_eq, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Mathlib.Tactic.Ring.instCommSemiringNat, Mathlib.Tactic.Ring.pow_congr, Mathlib.Tactic.Ring.atom_pf, Mathlib.Tactic.Ring.add_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.add_pf_add_gt, Mathlib.Tactic.Ring.add_pf_add_zero, Mathlib.Tactic.Ring.pow_add and 40 more
63 instances, coercions and auxiliary constants left out.

Tactic trace: 42 steps
Open to load the checked before-and-after goal states.
lemma p_pow_dvd_all_shifted line 523 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Nat.Prime, Ne, Eq, a, HAdd.hAdd, Dvd.dvd, Int, HPow.hPow, HSub.hSub, HMul.hMul
Proof uses: Nat.recAux, rfl, Eq.mpr, id, congrArg, p_sq_dvd_5n_4, Decidable.byContradiction, Not, p_pow_dvd_5n_4_step
34 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma p_pow_gt line 537 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, GT.gt, Int, HPow.hPow, HMul.hMul, HSub.hSub
Proof uses: LT.lt.trans, sub_lt_self, Int.NonNeg.mk, Nat.succ, cast, LT.lt, Eq.symm, Eq.trans, congr, congrArg, Nat.lt_pow_self, Decidable.byContradiction, Not
68 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma a_eq_p_implies_n_lt line 540 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Nat.Prime, Ne, Eq, a, LT.lt, HAdd.hAdd
Proof uses: Decidable.byContradiction, Not, Nat.Prime.two_le, Eq.mp, LE.le, Dvd.dvd, Int, HPow.hPow, HSub.hSub, HMul.hMul, p_pow_dvd_all_shifted, p_ge_7, Eq.mpr, id, congrArg, GT.gt, p_pow_gt, Int.le_of_dvd, False.elim, False and 58 more
90 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
def g_p line 557 source
no axioms what this means
Statement uses: Nat
Proof uses: ite, Eq, HMod.hMod, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
lemma g_p_prop line 564 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Ne, OfNat.ofNat, Eq, HMod.hMod, And, HMul.hMul, g_p, Membership.mem, Set, Insert.insert, Singleton.singleton
Proof uses: id, g_p_prop.match_1, ite, False, Function.comp, Dvd.dvd, Iff.mp, Nat.Prime.dvd_iff_eq, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.dvd_of_mod_eq_zero, of_eq_true, Eq.trans, True, congr, congrArg, ite_cond_eq_true and 9 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
def n_p line 569 source
no axioms what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, HAdd.hAdd, HMul.hMul, g_p, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
lemma n_p_prop line 571 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Ne, OfNat.ofNat, Eq, HMod.hMod, And, HSub.hSub, HMul.hMul, n_p, g_p, GE.ge
Proof uses: id, Decidable, HDiv.hDiv, HAdd.hAdd, ite, Not, Eq.mpr, congr, congrArg, if_neg, False, Iff.mp, Dvd.dvd, Nat.Prime.dvd_iff_eq, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Decidable.byContradiction and 3 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
lemma existence_intQ2_not_dvd line 581 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Ne, Not, Dvd.dvd, Int, IntQ2, n_p
Proof uses: id, Eq.mpr, HAdd.hAdd, HMul.hMul, HSub.hSub, HDiv.hDiv, g_p, sum_fact, Nat.factorial, Int.subNatNat, Eq.trans, congrArg, Eq.symm, Nat.cast_one, congr, ite, g_p.eq_1, Decidable, if_neg, absurd and 121 more
170 instances, coercions and auxiliary constants left out.

Tactic trace: 76 steps
Open to load the checked before-and-after goal states.
lemma a_3_eq_11 line 615 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, a, OfNat.ofNat
Proof uses: Eq.mpr, id, congrArg, Eq.symm, propext, eq_comm, ite, LT.lt, Rat.den, continued_fraction_val, a.eq_1, of_eq_true, Eq.trans, True, False, ite_congr, eq_false, Mathlib.Meta.NormNum.isNat_lt_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl and 35 more
57 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma n_p_ge_4 line 619 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Ne, GE.ge, n_p
Proof uses: id, Eq.mpr, HDiv.hDiv, HAdd.hAdd, HMul.hMul, g_p, LT.lt, ite, Eq.trans, HSub.hSub, of_eq_true, eq_true_of_decide, Eq.refl, Bool, Bool.true, congrArg, Nat.Simproc.add_sub_le, of_decide_eq_true, LE.le, n_p_ge_4.match_2 and 14 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma p_mod_5_cases line 625 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Ne, OfNat.ofNat, Or, Eq, HMod.hMod
Proof uses: Dvd.dvd, Nat.dvd_of_mod_eq_zero, Nat.Prime.eq_one_or_self_of_dvd, Or.inl, GT.gt, Nat.Prime.one_lt, Decidable.byContradiction, Not, Or.inr, Eq.symm, Eq.refl
13 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma two_g_p_dvd_intQ2 line 639 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Ne, Dvd.dvd, Int, HMul.hMul, g_p, IntQ2, n_p
Proof uses: Or, p_mod_5_cases, Eq.mpr, HAdd.hAdd, HSub.hSub, HDiv.hDiv, sum_fact, Nat.factorial, id, congr, ite, congrArg, Eq.trans, ite_cond_eq_true, True, eq_self, Nat.cast_one, mul_one, one_mul, Nat.cast_add and 143 more
184 instances, coercions and auxiliary constants left out.

Tactic trace: 81 steps
Open to load the checked before-and-after goal states.
lemma a_n_p_eq_p_of_ne_11 line 681 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Ne, a, n_p
Proof uses: GE.ge, n_p_ge_4, Rat.den, HDiv.hDiv, Rat, IntQ2, HMul.hMul, Int, HSub.hSub, a_val, And, g_p, n_p_prop, And.left, Decidable.byContradiction, Not, Eq.mpr, id, congrArg, Nat.cast_mul and 58 more
101 instances, coercions and auxiliary constants left out.

Tactic trace: 54 steps
Open to load the checked before-and-after goal states.
lemma uniqueness_part line 725 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Ne, GE.ge, a
Proof uses: False, False.elim, Eq.mp, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_natMod, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Eq.symm, Dvd.dvd, Int, HSub.hSub, HMul.hMul, a_eq_p_implies_p_divides, dvd_sub, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, HPow.hPow and 53 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
lemma existence_part line 742 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Ne, Exists, And, GE.ge, a
Proof uses: dite, Exists.intro, And.intro, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, a_3_eq_11, Eq.symm, Not, n_p, HSub.hSub, HMul.hMul, g_p, n_p_prop, And.right, a_n_p_eq_p_of_ne_11
21 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
theorem target_theorem_0 line 756 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, And, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Ne, ExistsUnique, GE.ge, a
Proof uses: Exists, existence_part, Exists.intro, And.intro, uniqueness_part
10 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_51293_conjecture_0.lean:517 (different declaration text), APNOutputs/OEIS/oeis_62567_conjecture_0.lean:329 (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: 7 steps
Open to load the checked before-and-after goal states.