`Compfiles/Usa2002P5.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 8 curated AI-written annotations and 3305 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned compfiles documentation, Pinned compfiles corpus index

16 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2002, Problem 5

Let a, b be integers greater than 2. Prove that there exists a positive integer k and a finite sequence n₁, n₂, ..., nₖ of positive integers such that n₁ = a, nₖ = b, and nᵢ + nᵢ₊₁ divides nᵢnᵢ₊₁ for each i (1 ≤ i < k).

Imports (8)

Imported by 1 file

Compfiles.lean

inductive Usa2002P5.Linked Linked a b means that a can be connected to b by a finite chain of positive integers in which any two consecutive terms x, y satisfy x + y ∣ x * y line 53 source
Source documentation
Linked a b means that a can be connected to b by a finite chain of positive integers in which any two consecutive terms x, y satisfy x + y ∣ x * y.

no axioms what this means
Statement uses: Int
```
inductive Linked : ℤ → ℤ → Prop where
…
```
theorem Usa2002P5.Linked.pos_left line 62 source
no axioms what this means
Statement uses: Int, Usa2002P5.Linked, LT.lt, OfNat.ofNat
Proof uses: Usa2002P5.Linked.rec, Dvd.dvd, HAdd.hAdd, HMul.hMul
9 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Usa2002P5.Linked.pos_right line 67 source
no axioms what this means
Statement uses: Int, Usa2002P5.Linked, LT.lt, OfNat.ofNat
Proof uses: Usa2002P5.Linked.rec, Dvd.dvd, HAdd.hAdd, HMul.hMul
9 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Usa2002P5.Linked.trans line 73 source
no axioms what this means
Statement uses: Int, Usa2002P5.Linked
Proof uses: Usa2002P5.Linked.rec, LT.lt, OfNat.ofNat, id, Dvd.dvd, HAdd.hAdd, HMul.hMul, Usa2002P5.Linked.snoc
7 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
instance Usa2002P5.Linked.(instance at line 79) line 79 source
theorem Usa2002P5.Linked.symm line 81 source
standard axioms: propext what this means
Statement uses: Int, Usa2002P5.Linked
Proof uses: Usa2002P5.Linked.rec, LT.lt, OfNat.ofNat, Usa2002P5.Linked.refl, Dvd.dvd, HAdd.hAdd, HMul.hMul, Usa2002P5.Linked.trans, Usa2002P5.Linked.snoc, Usa2002P5.Linked.pos_right, Eq.mpr, id, Eq, congrArg, add_comm, mul_comm
16 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Usa2002P5.link_pred_mul Basic link: t ~ t * (t - 1) for 1 < t, as t + t * (t - 1) = t ^ 2 line 91 source
Source documentation
Basic link: t ~ t * (t - 1) for 1 < t, as t + t * (t - 1) = t ^ 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LT.lt, OfNat.ofNat, Usa2002P5.Linked, HMul.hMul, HSub.hSub
Proof uses: mul_pos, Usa2002P5.Linked.snoc, Usa2002P5.Linked.refl, Exists.intro, Eq, HAdd.hAdd, Mathlib.Tactic.Ring.of_eq, HPow.hPow, Nat, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg and 34 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem link_pred_mul {t : ℤ} (ht : 1 < t) : Linked t (t * (t - 1)) := by
…
```
Mentions: Usa2002P5.Linked
theorem Usa2002P5.link_two_mul_pred Basic link: 2 * t ~ t * (t - 2) for 2 < t, as 2 * t + t * (t - 2) = t ^ 2 line 97 source
Source documentation
Basic link: 2 * t ~ t * (t - 2) for 2 < t, as 2 * t + t * (t - 2) = t ^ 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LT.lt, OfNat.ofNat, Usa2002P5.Linked, HMul.hMul, HSub.hSub
Proof uses: mul_pos, Usa2002P5.Linked.snoc, Usa2002P5.Linked.refl, Exists.intro, Eq, HAdd.hAdd, Mathlib.Tactic.Ring.of_eq, HPow.hPow, Nat, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.Common.atom_pf, rfl and 35 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem link_two_mul_pred {t : ℤ} (ht : 2 < t) : Linked (2 * t) (t * (t - 2)) := by
…
```
Mentions: Usa2002P5.Linked
theorem Usa2002P5.link_mul_left Scaling: a ~ b implies m * a ~ m * b for 0 < m line 103 source
Source documentation
Scaling: a ~ b implies m * a ~ m * b for 0 < m.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LT.lt, OfNat.ofNat, Usa2002P5.Linked, HMul.hMul
Proof uses: Usa2002P5.Linked.rec, Usa2002P5.Linked.refl, mul_pos, Dvd.dvd, HAdd.hAdd, Usa2002P5.Linked.snoc, Eq, Exists.intro, Mathlib.Tactic.LinearCombination.eq_of_eq, Mathlib.Tactic.LinearCombination.mul_const_eq, Mathlib.Tactic.LinearCombination.eq_rearrange, Mathlib.Tactic.Ring.of_eq, HSub.hSub, Mathlib.Tactic.Ring.Common.sub_congr, HPow.hPow, Nat, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf and 41 more
66 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem link_mul_left {m : ℤ} (hm : 0 < m) {a b : ℤ} (h : Linked a b) :
    Linked (m * a) (m * b) := by
…
```
Mentions: h, Usa2002P5.Linked
theorem Usa2002P5.link_two_mul Doubling: every 2 < t links to 2 * t line 113 source
Source documentation
Doubling: every 2 < t links to 2 * t.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LT.lt, OfNat.ofNat, Usa2002P5.Linked, HMul.hMul
Proof uses: HSub.hSub, Usa2002P5.link_pred_mul, Trans.trans, Eq, rfl, Usa2002P5.link_mul_left, Eq.mp, HAdd.hAdd, congrArg, Eq.symm, mul_assoc, Int.sub_sub, Usa2002P5.Linked.symm, Iff.mpr, Int.sub_pos, Int.mul_comm, mul_comm, Usa2002P5.link_two_mul_pred
25 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem link_two_mul {t : ℤ} (ht : 2 < t) : Linked t (2 * t) := by
…
```
Mentions: Usa2002P5.Linked
theorem Usa2002P5.link_pred Reduction: every 3 < t links to t - 1 line 127 source
Source documentation
Reduction: every 3 < t links to t - 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LT.lt, OfNat.ofNat, Usa2002P5.Linked, HSub.hSub
Proof uses: HMul.hMul, Usa2002P5.link_pred_mul, Eq.mp, congrArg, Eq, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, Int.rawCast, inferInstance, Ring, Int.negOfNat, HPow.hPow, Nat, Nat.rawCast, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, Eq.symm and 43 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 30 steps
Open to load the checked before-and-after goal states.
```
theorem link_pred {t : ℤ} (ht : 3 < t) : Linked t (t - 1) := by
…
```
Mentions: Usa2002P5.Linked
theorem Usa2002P5.linked_to_three Every integer ≥ 3 links to 3 line 156 source
Source documentation
Every integer ≥ 3 links to 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int, Eq, HAdd.hAdd, OfNat.ofNat, Usa2002P5.Linked
Proof uses: Nat.recAux, Usa2002P5.Linked.refl, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_natCast, Bool, Bool.false, Eq.symm, Usa2002P5.Linked.trans, HSub.hSub, Usa2002P5.link_pred
31 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem linked_to_three : ∀ n : ℕ, ∀ t : ℤ, t = n + 3 → Linked t 3 := by
…
```
Mentions: Usa2002P5.Linked
theorem Usa2002P5.linked_of_two_lt line 166 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LT.lt, OfNat.ofNat, Usa2002P5.Linked
Proof uses: Usa2002P5.linked_to_three, Int.toNat, HSub.hSub, Eq, Int.toNat_of_nonneg
10 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Usa2002P5.linked_any Any two integers greater than 2 are linked (via 3) line 172 source
Source documentation
Any two integers greater than 2 are linked (via 3).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LT.lt, OfNat.ofNat, Usa2002P5.Linked
Proof uses: Usa2002P5.Linked.trans, Usa2002P5.linked_of_two_lt, Usa2002P5.Linked.symm
4 instances, coercions and auxiliary constants left out.
```
theorem linked_any {a b : ℤ} (ha : 2 < a) (hb : 2 < b) : Linked a b :=
…
```
Mentions: Usa2002P5.Linked
theorem Usa2002P5.exists_seq_of_linked Unfold Linked into an explicit sequence line 176 source
Source documentation
Unfold Linked into an explicit sequence.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Usa2002P5.Linked, Exists, Nat, And, Eq, OfNat.ofNat, LE.le, LT.lt, Dvd.dvd, HAdd.hAdd, HMul.hMul
Proof uses: Usa2002P5.Linked.rec, Exists.intro, And.intro, rfl, Nat.eq_zero_of_le_zero, Eq.symm, absurd, Nat.not_lt_zero, Function.update, Eq.mpr, id, congrArg, Function.update_of_ne, Function.update_self, Eq.refl, dite, Not
33 instances, coercions and auxiliary constants left out.

Tactic trace: 40 steps
Open to load the checked before-and-after goal states.
```
theorem exists_seq_of_linked {a b : ℤ} (h : Linked a b) :
    ∃ k : ℕ, ∃ n : ℕ → ℤ, n 0 = a ∧ n k = b ∧
      (∀ i, i ≤ k → 0 < n i) ∧ ∀ i, i < k → n i + n (i + 1) ∣ n i * n (i + 1) := by
…
```
Mentions: h, Usa2002P5.Linked
problem Usa2002P5.usa2002_p5 line 208 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LT.lt, OfNat.ofNat, Exists, Nat, And, Eq, LE.le, Dvd.dvd, HAdd.hAdd, HMul.hMul
Proof uses: Usa2002P5.exists_seq_of_linked, Usa2002P5.linked_any
13 instances, coercions and auxiliary constants left out.