Squarefree.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

19 declarations, 0 contain sorry, 0 declared as axioms

Imports (4)

Imported by 1 file

FormalConjecturesForMathlib.lean

def Nat.squarefreePart n.squarefreePart is the unique a₀ : ℕ such that a₀ is squarefree and n = a₀ * b ^ 2, for some b : ℕ line 33 source
Source documentation
n.squarefreePart is the unique a₀ : ℕ such that a₀ is squarefree and n = a₀ * b ^ 2, for some b : ℕ. At 0 this property is not well defined because b must be 0 and a₀ can be any squarefree number; we give the junk value 1 at n = 0 following the convention that the squarefree part of any square is 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finsupp.prod, Nat.factorization, HPow.hPow, HMod.hMod, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
def squarefreePart (n : ℕ) : ℕ := n.factorization.prod fun (p e : ℕ) ↦ p ^ (e % 2)
```
example Nat.(example at line 35) line 35 source
example Nat.(example at line 38) line 38 source
example Nat.(example at line 41) line 41 source
example Nat.(example at line 44) line 44 source
example Nat.(example at line 47) line 47 source
example Nat.(example at line 50) line 50 source
theorem Nat.squarefreePart_ne_zero line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Ne, Nat.squarefreePart, OfNat.ofNat
Proof uses: of_eq_true, Not, Eq, Finsupp.prod, Nat.factorization, HPow.hPow, HMod.hMod, Eq.trans, False, True, congrArg, Exists, And, Membership.mem, Finset, Finsupp.support, Finsupp, Nat.Prime, Dvd.dvd, funext and 16 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Nat.squarefreePart_zero line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.squarefreePart, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Finsupp.prod, Nat.factorization, Finsupp, Nat.factorization_zero, HPow.hPow, HMod.hMod, eq_self
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Nat.squarefreePart_of_squarefree If n is squarefree, then its squarefree part is itself line 60 source
Source documentation
If n is squarefree, then its squarefree part is itself.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Squarefree, Eq, Nat.squarefreePart
Proof uses: Eq.mpr, Finsupp.prod, Nat.factorization, HPow.hPow, id, congrArg, Eq.symm, Nat.prod_factorization_pow_eq_self, OfNat.ofNat, False.elim, False, Eq.mp, Eq.trans, Finset.prod_congr, Nat.primeFactors, HMod.hMod, Finsupp, rfl, Finset, Membership.mem and 10 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem squarefreePart_of_squarefree {n : ℕ} (hn : Squarefree n) :
    squarefreePart n = n := by
…
```
Mentions: Nat.squarefreePart
theorem Nat.squarefreePart_of_isSquare The squarefree part of any square is 1 line 68 source
Source documentation
The squarefree part of any square is 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, IsSquare, Eq, Nat.squarefreePart, OfNat.ofNat
Proof uses: Ne, Or, eq_or_ne, Nat.squarefreePart_zero, Eq.symm, HMul.hMul, Eq.mpr, Finsupp.prod, Nat.factorization, HPow.hPow, HMod.hMod, id, congrArg, Nat.squarefreePart.eq_1, Finset.prod, Finsupp.support, Finsupp, Finsupp.prod.eq_1, Nat.primeFactors, Finset and 21 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem squarefreePart_of_isSquare {n : ℕ} (hn : IsSquare n) :
    squarefreePart n = 1 := by
…
```
Mentions: Nat.squarefreePart
theorem Nat.squarefreePart_factorization If $n = \prod_p p^{e_p}$ is the prime factorization of $n$, then $\prod_p p^{e_p \pmod{2}}$ is the prime factorization of the squarefree part of $n$ line 79 source
Source documentation
If $n = \prod_p p^{e_p}$ is the prime factorization of $n$, then $\prod_p p^{e_p \pmod{2}}$ is the prime factorization of the squarefree part of $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Eq, Finsupp, Nat.factorization, Nat.squarefreePart, HMod.hMod, OfNat.ofNat
Proof uses: Eq.mpr, Finsupp.prod, HPow.hPow, id, congrArg, Nat.squarefreePart.eq_1, Finset.prod, Nat.primeFactors, Nat.prod_factorization_eq_prod_primeFactors, Nat.prod_primeFactors_factorization_apply, Not, Dvd.dvd, of_eq_true, Eq.trans, True, congrFun', Nat.factorization_eq_zero_of_not_dvd, eq_self, forall_congr, Nat.factorization_zero and 2 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem squarefreePart_factorization (n : ℕ) {p : ℕ} (hp : p.Prime) :
    n.squarefreePart.factorization p = n.factorization p % 2 := by
…
```
theorem Nat.Prime.squarefree line 85 source
standard axioms: propext what this means
Statement uses: Nat, Nat.Prime, Squarefree
Proof uses: Irreducible.squarefree
2 instances, coercions and auxiliary constants left out.
theorem Nat.squarefree_squarefreePart line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Squarefree, Nat.squarefreePart
Proof uses: Iff.mpr, LE.le, Finsupp, Nat.factorization, OfNat.ofNat, Nat.squarefree_iff_factorization_le_one, Nat.squarefreePart_ne_zero, dite, Nat.Prime, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, HAdd.hAdd, Neg.neg, HSub.hSub, HMod.hMod, congrArg, Mathlib.Tactic.Ring.of_eq and 68 more
92 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem Nat.squarefreePart_dvd line 93 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Dvd.dvd, Nat.squarefreePart
Proof uses: Eq, OfNat.ofNat, Ne, Or, eq_or_ne, of_eq_true, Eq.symm, Iff.mp, Nat.Prime, LE.le, Finsupp, Nat.factorization, Nat.factorization_prime_le_iff_dvd, Nat.squarefreePart_ne_zero, Eq.rec, HMod.hMod, Nat.mod_le, Nat.squarefreePart_factorization
18 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def Nat.squarePart The square part is the value of b ^ 2 in the squarefree decomposition of n = a₀ * b ^ 2 line 99 source
Source documentation
The square part is the value of b ^ 2 in the squarefree decomposition of n = a₀ * b ^ 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, Nat.squarefreePart
2 instances, coercions and auxiliary constants left out.
```
def squarePart (n : ℕ) : ℕ := n / n.squarefreePart
```
theorem Nat.squarePart_zero line 101 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.squarePart, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.zero_div, Nat.squarefreePart, eq_self
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Nat.squarefreePart_mul_squarePart The squarefree decomposition of a natural number line 104 source
Source documentation
The squarefree decomposition of a natural number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, HMul.hMul, Nat.squarefreePart, Nat.squarePart
Proof uses: Iff.mpr, HDiv.hDiv, Dvd.dvd, Nat.mul_div_eq_iff_dvd, Nat.squarefreePart_dvd
7 instances, coercions and auxiliary constants left out.
```
theorem squarefreePart_mul_squarePart (n : ℕ) : n.squarefreePart * n.squarePart = n :=
…
```
theorem Nat.squarefree_infinite line 107 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, Squarefree
Proof uses: Set.Infinite.mono, Nat.Prime, Membership.mem, Set, Nat.Prime.squarefree, Nat.infinite_setOfPred_prime
3 instances, coercions and auxiliary constants left out.