PerfectPower.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 10 curated AI-written annotations and 7190 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 project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 0 contain sorry, 0 declared as axioms

Imports (4)

Imported by 1 file

FormalConjecturesForMathlib.lean

def Nat.IsPerfectPower Documented declaration: IsPerfectPower line 36 source
Source documentation
A perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, $n$ is a perfect power if there exist natural numbers $m > 1$, and $k > 1$ such that $m ^ k = n$. In this case, $n$ may be called a perfect $k$th power. If $k = 2$ or $k = 3$, then $n$ is called a perfect square or perfect cube, respectively.

L4DEL explanation
A perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, $n$ is a perfect power if there exist natural numbers $m > 1$, and $k > 1$ such that $m ^ k = n$. In this case, $n$ may be called a perfect $k$th power. If $k = 2$ or $k = 3$, then $n$ is called a perfect square or perfect cube, respectively.
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, And, LT.lt, OfNat.ofNat, Eq, HPow.hPow
6 instances, coercions and auxiliary constants left out.
```
def IsPerfectPower (n : ℕ) : Prop :=
…
```
theorem Nat.isPerfectPower_iff_factorization_gcd line 39 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, Nat.IsPerfectPower, And, GT.gt, OfNat.ofNat, Finset.gcd, Nat.primeFactors, Finsupp, Nat.factorization
Proof uses: Iff.intro, LT.lt, Eq, HPow.hPow, And.intro, GE.ge, Trans.trans, Eq.symm, Nat.pow_le_pow_left, Nat.succ, Nat.pow_le_pow_right, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Decidable.byContradiction, Not, Ne, Iff.mp and 63 more
71 instances, coercions and auxiliary constants left out.

Tactic trace: 81 steps
Open to load the checked before-and-after goal states.
instance Nat.IsPerfectPower.decide line 129 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Decidable, Nat.IsPerfectPower
Proof uses: decidable_of_iff, And, GT.gt, OfNat.ofNat, Finset.gcd, Nat.primeFactors, Finsupp, Nat.factorization
12 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
example Nat.(example at line 133) line 133 source
example Nat.(example at line 134) line 134 source
example Nat.(example at line 135) line 135 source
example Nat.(example at line 136) line 136 source
example Nat.(example at line 137) line 137 source