lean_workbook_plus_5972

⚠️ Retired — specification defect

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

If $f(x)^2 = g(x)^2$ for all $x$ in an interval $I$, what is a sufficient condition to deduce that $f(x) = g(x)$ or $f(x) = -g(x)$ for all $x$ in $I$?

Why this node was retired

The posted statement is

theorem lean_workbook_plus_5972 (f g : ℝ → ℝ) (I : Set ℝ) (hI : I ≠ ∅) (h : f^2 = g^2) : ∀ x ∈ I, f x = g x ∨ f x = -g x   :=  by sorry

The target moves the disjunction inside ∀x so the sign may vary independently at each point; it does not answer the global-sign question.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

For an interval I, assume f,g continuous on I, g has no zeros on I, and f²=g² on I; conclude either f=g throughout I or f=−g throughout I. These are concrete sufficient conditions, as requested.

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.