Subfield_Test

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$. Let $K$ be a subset of $F$. Then $\struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$ these all hold $(1): \quad K^* \ne \O$ $(2): \quad \forall x, y \in K: x + \paren {-y} \in K$ $(3): \quad \forall x, y \in K: x \circ y \in K$ $(4): \quad x \in K^* \implies x^{-1} \in K^*$ where $K^*$ denotes $K \setminus \set {0_F}$.

Why this node was retired

The posted statement is

theorem Subfield_Test {F : Type _} [Field F] (K : Set F) (hne : K.Nonempty) (hsub : ∀ a b : F, a ∈ K → b ∈ K → a - b ∈ K) (hmul : ∀ a b : F, a ∈ K → b ∈ K → b ≠ 0 → a * b⁻¹ ∈ K) : ∃ S : Subfield F, (↑S : Set F) = K := by sorry

Nonempty K replaces existence of a nonzero element of K.

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 ∃x∈K,x≠0, closure under subtraction, and closure under a/b when a,b∈K and b≠0. These conditions are equivalent to K being the carrier of a subfield.

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.