`Compfiles/Usa2006P4.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

7 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2006, Problem 4

Find all positive integers n for which there exist an integer k ≥ 2 and positive rational numbers a₁, a₂, ..., aₖ satisfying a₁ + a₂ + ... + aₖ = a₁ · a₂ · ... · aₖ = n.

Imports (6)

Imported by 1 file

Compfiles.lean

determine Usa2006P4.SolutionSet line 32 source
no axioms what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Or, Eq, OfNat.ofNat, LE.le
2 instances, coercions and auxiliary constants left out.
lemma Usa2006P4.not_isSquare_five_nat 5 is not the square of a natural number line 49 source
Source documentation
5 is not the square of a natural number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, IsSquare, Nat, OfNat.ofNat
Proof uses: Eq, HMul.hMul, False, LE.le, Decidable.byContradiction, Nat.mul_le_mul, dite, False.elim, Eq.mp, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_mul, Bool, Bool.false, Eq.symm, le_antisymm, Nat.rawCast, Mathlib.Tactic.IntervalCases.of_le_right and 3 more
17 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma not_isSquare_five_nat : ¬ IsSquare (5 : ℕ) := by
…
```
lemma Usa2006P4.rat_mul_self_ne_five 5 is not the square of a rational number line 59 source
Source documentation
5 is not the square of a rational number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Ne, HMul.hMul, OfNat.ofNat
Proof uses: Eq, Irrational, Real.sqrt, Real, Nat, Nat.Prime.irrational_sqrt, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4, Mathlib.Meta.NormNum.IsNat.raw_refl, Mathlib.Meta.NormNum.minFacHelper_0, Bool.false, Exists.intro, abs, HPow.hPow, cast, Eq.symm and 33 more
60 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
lemma rat_mul_self_ne_five (q : ℚ) : q * q ≠ 5 := by
…
```
lemma Usa2006P4.sum_eq_sum_get Summing the entries of a list of rationals via Fin-indexed access line 72 source
Source documentation
Summing the entries of a list of rationals via Fin-indexed access.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Rat, Eq, Finset.sum, Fin, List.length, Finset.univ, List.get, List.sum
Proof uses: Eq.mpr, List.ofFn, id, congrArg, Eq.symm, Fin.sum_ofFn, List.ofFn_get, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma sum_eq_sum_get (l : List ℚ) : ∑ i : Fin l.length, l.get i = l.sum := by
…
```
lemma Usa2006P4.prod_eq_prod_get Multiplying the entries of a list of rationals via Fin-indexed access line 76 source
Source documentation
Multiplying the entries of a list of rationals via Fin-indexed access.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Rat, Eq, Finset.prod, Fin, List.length, Finset.univ, List.get, List.prod
Proof uses: Eq.mpr, List.ofFn, id, congrArg, Eq.symm, Fin.prod_ofFn, List.ofFn_get, Eq.refl
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma prod_eq_prod_get (l : List ℚ) : ∏ i : Fin l.length, l.get i = l.prod := by
…
```
lemma Usa2006P4.prod_le_sum_div_pow AM–GM, raised to the k-th power, for positive rationals indexed by Fin k: the product is at most the k-th power of the arithmetic mean line 81 source
Source documentation
AM–GM, raised to the k-th power, for positive rationals indexed by Fin k: the product is at most the k-th power of the arithmetic mean.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Fin, Rat, LE.le, Real, Finset.prod, Finset.univ, HPow.hPow, HDiv.hDiv, Finset.sum
Proof uses: Ne, cast, Not, Eq, Eq.symm, Eq.trans, congrArg, Nat.cast_zero, LT.lt.ne', Iff.mpr, Rat.cast_nonneg, LT.lt.le, Inv.inv, HMul.hMul, Real.geom_mean_le_arith_mean_weighted, Membership.mem, Finset, inv_nonneg, Nat.cast_nonneg, Eq.mpr and 25 more
140 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
lemma prod_le_sum_div_pow {k : ℕ} (hk : 0 < k) (z : Fin k → ℚ) (hz : ∀ i, 0 < z i) :
    ((∏ i, z i : ℚ) : ℝ) ≤ (((∑ i, z i : ℚ) : ℝ) / k) ^ k := by
…
```
problem Usa2006P4.usa2006_p4 line 106 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Membership.mem, Set, Usa2006P4.SolutionSet, Exists, List, Rat, And, LE.le, List.length, Eq, List.sum, List.prod
Proof uses: id, Iff.intro, Or, Exists.intro, List.cons, List.nil, And.intro, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mp, Eq.trans, congrArg, False, or_false, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Bool.false and 174 more
242 instances, coercions and auxiliary constants left out.

Tactic trace: 160 steps
Open to load the checked before-and-after goal states.