Ring_of_Sets_Closed_under_Finite_Intersection
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.
A ring of sets is closed under finite intersection.
Why this node was retired
The posted statement is
theorem Ring_of_Sets_Closed_under_Finite_Intersection {α : Type _} (R : Set (Set α)) (hR_empty : ∅ ∈ R) (hR_union : ∀ A B : Set α, A ∈ R → B ∈ R → A ∪ B ∈ R) (hR_diff : ∀ A B : Set α, A ∈ R → B ∈ R → A \ B ∈ R) (n : ℕ) (A : Fin n → Set α) (hA : ∀ i, A i ∈ R) : (⋂ i, A i) ∈ R := by sorry
The assertion includes the empty intersection without requiring the universal set in the ring of sets.
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 0<n for closure under intersections indexed by Fin n, or add Set.univ∈R if empty intersections are included.
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.Analysis.Complex.Basic
theorem Ring_of_Sets_Closed_under_Finite_Intersection {α : Type _} (R : Set (Set α)) (hR_empty : ∅ ∈ R) (hR_union : ∀ A B : Set α, A ∈ R → B ∈ R → A ∪ B ∈ R) (hR_diff : ∀ A B : Set α, A ∈ R → B ∈ R → A \ B ∈ R) (n : ℕ) (A : Fin n → Set α) (hA : ∀ i, A i ∈ R) : (⋂ i, A i) ∈ R := by sorry