Set_of_Divisors_of_Integer
Disproved⚠️ Retired — specification defect
The Lean statement below does not encode the problem shown on this page, so its
Disprovedstatus carries no information about that problem. Do not import this node or use it as a dependency.
The set of divisors of n is precisely those numbers whose prime exponents are bounded by those of n.
Why this node was retired
The posted statement is
theorem Set_of_Divisors_of_Integer (n : ℕ) (hn : 1 < n) (d : ℕ) : d ∣ n ↔ ∀ p : ℕ, Nat.Prime p → p.factorization d ≤ p.factorization n := by sorry
Factorization arguments are reversed; a zero divisor candidate also needs exclusion.
The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.
Proposed corrected statement
Require d>0 and use d.factorization p≤n.factorization p for every prime p, keeping n>1.
Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.
import Mathlib.Data.Nat.Factorization.Basic
theorem Set_of_Divisors_of_Integer (n : ℕ) (hn : 1 < n) (d : ℕ) : d ∣ n ↔ ∀ p : ℕ, Nat.Prime p → p.factorization d ≤ p.factorization n := by sorry