erdos_741.parts.ii.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

39 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

FormalConjectures.Util.ProblemImports

lemma Erdos741.basis_of_order_two_iff line 79 source
standard axioms: propext what this means
Statement uses: Set, Nat, Iff, Set.IsAddBasisOfOrder, Union.union, Singleton.singleton, OfNat.ofNat, Membership.mem, HSMul.hSMul
Proof uses: Iff.rfl
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.answer_true_iff line 82 source
no axioms what this means
Statement uses: Iff, True
Proof uses: Iff.rfl

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.miss_gap line 85 source
no axioms what this means
Statement uses: Set, Nat, HasSubset.Subset, SDiff.sdiff, HAdd.hAdd, Compl.compl
Proof uses: Membership.mem, And.right, Not
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.not_syndetic_of_large_gaps line 90 source
no axioms what this means
Statement uses: Set, Nat, Exists, HasSubset.Subset, Set.Icc, HAdd.hAdd, Compl.compl, Not, IsSyndetic
Proof uses: Set.Nonempty, Inter.inter, False, Membership.mem, And.left, And.right
13 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
def Erdos741.GoodCasselsProperty line 104 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: And, Set.IsAddBasisOfOrder, Union.union, Singleton.singleton, OfNat.ofNat, Eq, Disjoint, Or, Exists, HasSubset.Subset, Set.Icc, HAdd.hAdd, Compl.compl
19 instances, coercions and auxiliary constants left out.
def Erdos741.BlockSeq line 110 source
no axioms what this means
Proof uses: Nat, Set
def Erdos741.UnionBlocks line 112 source
no axioms what this means
Statement uses: Erdos741.BlockSeq, Set, Nat
Proof uses: Set.iUnion
lemma Erdos741.subset_union_blocks line 114 source
standard axioms: propext what this means
Statement uses: Erdos741.BlockSeq, Nat, HasSubset.Subset, Set, Erdos741.UnionBlocks
Proof uses: Membership.mem, Eq.mpr, Exists, id, Eq, Exists.intro
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.sum_subset_union_sum line 119 source
standard axioms: propext what this means
Statement uses: Erdos741.BlockSeq, Nat, HasSubset.Subset, Set, HAdd.hAdd, Erdos741.UnionBlocks
Proof uses: Membership.mem, And, Exists, Eq, Exists.intro, And.intro, Erdos741.subset_union_blocks
10 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.add_zero_subset line 130 source
no axioms what this means
Statement uses: Set, Nat, HasSubset.Subset, HAdd.hAdd, Union.union, Singleton.singleton, OfNat.ofNat
Proof uses: Membership.mem, And, Exists, Eq, Exists.intro, And.intro, Or.inl
16 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.icc_nonempty_custom line 141 source
no axioms what this means
Statement uses: Nat, Set.Nonempty, Set.Icc, HAdd.hAdd
Proof uses: Exists.intro, Membership.mem, Set, And.intro, LE.le, le_rfl, Nat.le_add_right
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.icc_subset_complement line 145 source
no axioms what this means
Statement uses: Set, Nat, Membership.mem, Set.Icc, HAdd.hAdd, Not, HasSubset.Subset, Compl.compl
10 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def Erdos741.HasLargeGaps line 149 source
no axioms what this means
Statement uses: Set, Nat
Proof uses: Exists, LE.le, HAdd.hAdd, Not, Membership.mem
4 instances, coercions and auxiliary constants left out.
lemma Erdos741.syndetic_not_large_gaps line 152 source
no axioms what this means
Statement uses: Set, Nat, IsSyndetic, Not, Erdos741.HasLargeGaps
Proof uses: Set.Nonempty, Inter.inter, Set.Icc, HAdd.hAdd, Exists, False, LE.le, Membership.mem, And.left, And.right
8 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.has_gaps_mono line 165 source
no axioms what this means
Statement uses: Set, Nat, LE.le, Exists, HAdd.hAdd, Not, Membership.mem
Proof uses: Exists.intro, LE.le.trans, Nat.add_le_add_left
10 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.infinite_or line 173 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, Or
Proof uses: dite, Or.inl, Not, Or.inr, Exists, Eq.mp, Mathlib.Tactic.Push.not_forall_eq, Eq, False.elim, Eq.refl, le_of_lt, LT.lt
12 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
def Erdos741.IsGreedyBasis line 198 source
no axioms what this means
Statement uses: Nat, Set
Proof uses: Exists, And, Membership.mem, Union.union, Singleton.singleton, OfNat.ofNat, Eq, HAdd.hAdd
6 instances, coercions and auxiliary constants left out.
def Erdos741.GreedySpaced line 201 source
no axioms what this means
Statement uses: Nat, Set
Proof uses: And, HasSubset.Subset, HAdd.hAdd, OfNat.ofNat, LE.le, Membership.mem, SDiff.sdiff, GT.gt
8 instances, coercions and auxiliary constants left out.
def Erdos741.GreedyGaps line 206 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Exists, And, LE.le, HAdd.hAdd, Eq, Union.union, Disjoint, Or, Not, Membership.mem
15 instances, coercions and auxiliary constants left out.
def Erdos741.State line 211 source
no axioms what this means
Proof uses: Subtype, Prod, Set, Nat, Membership.mem, Prod.fst, LE.le, Prod.snd
2 instances, coercions and auxiliary constants left out.
def Erdos741.step_prop line 213 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos741.State, Nat
Proof uses: And, HasSubset.Subset, Set, Prod.fst, Prod, Membership.mem, LE.le, Prod.snd, SDiff.sdiff, GT.gt, Exists, HAdd.hAdd, Eq, Union.union, Disjoint, Or, Not, Singleton.singleton, OfNat.ofNat
21 instances, coercions and auxiliary constants left out.
lemma Erdos741.valid_ext_exists line 222 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos741.State, Nat, Exists, Erdos741.step_prop
Proof uses: Prod.snd, Set, Prod, Membership.mem, Prod.fst, LE.le, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Union.union, Set.Icc, Singleton.singleton, Or, And, Eq, Eq.mp, Eq.trans, congr, congrArg, Subtype.property and 29 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 270 steps (first 200 of 270)
Open to load the checked before-and-after goal states.
def Erdos741.seq_step line 392 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Erdos741.State
Proof uses: Erdos741.seq_step.match_1, Unit, OfNat.ofNat, Subtype.mk, Prod, Set, Membership.mem, Prod.fst, LE.le, Prod.snd, Prod.mk, EmptyCollection.emptyCollection, Nat.succ, Classical.choose, Erdos741.step_prop
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def Erdos741.f_seq line 396 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Prod.fst, Prod, Membership.mem, LE.le, Prod.snd, Erdos741.seq_step
3 instances, coercions and auxiliary constants left out.
def Erdos741.gap_seq line 397 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Prod.snd, Set, Prod, Membership.mem, Prod.fst, LE.le, Erdos741.seq_step
3 instances, coercions and auxiliary constants left out.
lemma Erdos741.seq_step_prop line 399 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Erdos741.step_prop, Erdos741.seq_step, HAdd.hAdd, OfNat.ofNat
Proof uses: Classical.choose_spec, Erdos741.State, Erdos741.valid_ext_exists
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.f_seq_covers line 402 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, Erdos741.gap_seq, Exists, And, Membership.mem, Set, Union.union, Erdos741.f_seq, Singleton.singleton, OfNat.ofNat, Eq, HAdd.hAdd
Proof uses: Nat.recAux, Nat.eq_zero_of_le_zero, Exists.intro, Or.inr, Set.mem_singleton, And.intro, Eq.mpr, id, congrArg, Eq.refl, Erdos741.step_prop, Erdos741.seq_step, Erdos741.seq_step_prop, HasSubset.Subset, Prod.fst, Prod, Prod.snd, SDiff.sdiff, GT.gt, Disjoint and 2 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.f_seq_basis line 416 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos741.IsGreedyBasis, Erdos741.f_seq
Proof uses: Nat, Erdos741.step_prop, Erdos741.seq_step, HAdd.hAdd, OfNat.ofNat, Erdos741.seq_step_prop, HasSubset.Subset, Set, Prod.fst, Prod, Membership.mem, LE.le, Prod.snd, And, SDiff.sdiff, GT.gt, Exists, Eq, Union.union, Disjoint and 60 more
91 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.f_seq_spaced line 428 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos741.GreedySpaced, Erdos741.f_seq, Erdos741.gap_seq
Proof uses: And.intro, Nat, HasSubset.Subset, Set, HAdd.hAdd, OfNat.ofNat, And, LE.le, Membership.mem, SDiff.sdiff, GT.gt, Erdos741.step_prop, Erdos741.seq_step, Erdos741.seq_step_prop, And.left, Prod.fst, Prod, Prod.snd, Exists, Eq and 6 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.f_seq_gaps line 441 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos741.GreedyGaps, Erdos741.f_seq, Erdos741.gap_seq
Proof uses: Nat, Exists.intro, Exists, And, LE.le, HAdd.hAdd, Set, Eq, Union.union, Disjoint, Or, Not, Membership.mem, OfNat.ofNat, Erdos741.step_prop, Erdos741.seq_step, Erdos741.seq_step_prop, And.left, Prod.snd, Prod and 6 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.greedy_seq_exists line 447 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Nat, Set, And, Erdos741.IsGreedyBasis, Erdos741.GreedySpaced, Erdos741.GreedyGaps
Proof uses: Exists.intro, Erdos741.f_seq, Erdos741.gap_seq, And.intro, Erdos741.f_seq_basis, Erdos741.f_seq_spaced, Erdos741.f_seq_gaps

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.f_mono line 452 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Set, Erdos741.GreedySpaced, LE.le, HasSubset.Subset
Proof uses: Nat.le.rec, subset_refl, Nat.le, HasSubset.Subset.trans, Nat.succ, And.left, HAdd.hAdd, OfNat.ofNat, And, Membership.mem, SDiff.sdiff, GT.gt
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.gap_mono line 457 source
no axioms what this means
Statement uses: Nat, Set, Erdos741.GreedySpaced, LE.le
Proof uses: Nat.le.rec, le_rfl, Nat.le, LE.le.trans, Nat.succ, And.left, HAdd.hAdd, OfNat.ofNat, Membership.mem, SDiff.sdiff, GT.gt, And.right, HasSubset.Subset, And
10 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.subset_f_k_of_le line 462 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Erdos741.GreedySpaced, Membership.mem, Set.iUnion, LE.le
Proof uses: Exists, Iff.mp, Set.mem_iUnion, dite, Not, False.elim, And, LT.lt, Exists.intro, Nat.find, And.intro, Nat.find_spec, Nat.find_min, False, GT.gt, Decidable.byContradiction, le_of_not_gt, Mathlib.Tactic.Linarith.lt_irrefl, Int, OfNat.ofNat and 61 more
92 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.erdos_gap_set_exists line 496 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Set, Nat, And, Set.IsAddBasisOfOrder, Union.union, Singleton.singleton, OfNat.ofNat, Eq, Disjoint, Or, Erdos741.HasLargeGaps, HAdd.hAdd
Proof uses: Erdos741.IsGreedyBasis, Erdos741.GreedySpaced, Erdos741.GreedyGaps, Erdos741.greedy_seq_exists, Set.iUnion, Exists.intro, And.intro, Eq.mpr, Membership.mem, HSMul.hSMul, id, congrArg, propext, Erdos741.basis_of_order_two_iff, Or.inl, Erdos741.subset_union_blocks, Or.inr, Eq.refl, Eq.symm, two_nsmul and 69 more
129 instances, coercions and auxiliary constants left out.

Tactic trace: 125 steps
Open to load the checked before-and-after goal states.
lemma Erdos741.exists_good_cassels_set line 593 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Set, Nat, Erdos741.GoodCasselsProperty
Proof uses: And, Set.IsAddBasisOfOrder, Union.union, Singleton.singleton, OfNat.ofNat, Eq, Disjoint, Or, Erdos741.HasLargeGaps, HAdd.hAdd, Erdos741.erdos_gap_set_exists, Exists.intro, And.intro, HasSubset.Subset, Set.Icc, Compl.compl, Or.inl, LE.le, Not, Membership.mem and 5 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 30 steps
Open to load the checked before-and-after goal states.
def Erdos741.cassels_set line 625 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Classical.choose, Erdos741.GoodCasselsProperty, Erdos741.exists_good_cassels_set
lemma Erdos741.cassels_set_is_good line 628 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos741.GoodCasselsProperty, Erdos741.cassels_set
Proof uses: Classical.choose_spec, Set, Nat, Erdos741.exists_good_cassels_set
theorem Erdos741.target_theorem_0 Erdős problem 741(ii): an order-two basis with no syndetic two-part split line 634 source
L4DEL explanation
We show that the answer is yes by constructing a pathological basis A with “forbidden zones”
— Human-written proof companion for Erdős 741(ii)
There exists an additive basis A ∪ {0} of order two such that every disjoint partition A=A₁∪A₂ leaves at least one of A₁+A₁ and A₂+A₂ with arbitrarily large gaps.
1. Build A from dense clumps separated by rapidly growing forbidden zones, retaining one specially placed oasis in each zone.
2. Every natural number is still a sum of two elements of A ∪ {0}, either directly, by halving, or by using the oasis.
3. In any two-colouring of A, the oasis has one colour; a long interval of sums then becomes unavailable to the other colour, destroying syndeticity.
Sources and attribution: Pinned Lean source (0647711a7118), Human-written proof companion for Erdős 741(ii), Erdős Problems 741 source record
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Set.IsAddBasisOfOrder, Union.union, Singleton.singleton, OfNat.ofNat, Eq, Disjoint, Not, IsSyndetic, HAdd.hAdd
Proof uses: Eq.mpr, id, congrArg, propext, Erdos741.answer_true_iff, Iff.intro, Exists.intro, Erdos741.cassels_set, Erdos741.GoodCasselsProperty, Erdos741.cassels_set_is_good, Or, HasSubset.Subset, Set.Icc, Compl.compl, And.intro, False, Or.inl, Erdos741.not_syndetic_of_large_gaps, False.elim, Or.inr and 2 more
37 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/ErdosProblems/erdos_741.parts.i.lean:1486 (different declaration text). Qualify references as Erdos741.target_theorem_0@path.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
```
theorem target_theorem_0
  : answer(True) ↔ ∃ A : Set ℕ, IsAddBasisOfOrder (A ∪ {0}) 2 ∧ ∀ A₁ A₂, A = A₁ ∪ A₂ → Disjoint A₁ A₂ → ¬(IsSyndetic (A₁ + A₁) ∧ IsSyndetic (A₂ + A₂)) := by
…
```