lean_workbook_plus_53533

⚠️ 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.

Defining $f(a,b,c)=\frac{a+3c}{a+2b+c}+\frac{4b}{a+b+2c}-\frac{8c}{a+b+3c}$ you must solve the system \n\n $f_a=0$ \n\n $f_b=0$ \n\n $f_c=0$ \n\n this means \n\n $\left( a+2\,b+c \right) ^{-1}-{\frac {a+3\,c}{ \left( a+2\,b+c \right) ^{2}}}-4\,{\frac {b}{ \left( a+b+2\,c \right) ^{2}}}+8\,{\frac {c}{ \left( a+b+3\,c \right) ^{2}}}=0$ \n\n $-2\,{\frac {a+3\,c}{ \left( a+2\,b+c \right) ^{2}}}+4\, \left( a+b+2\,c \right) ^{-1}-4\,{\frac {b}{ \left( a+b+2\,c \right) ^{2}}}+8\,{\frac {c}{ \left( a+b+3\,c \right) ^{2}}}=0$ \n\n $3\, \left( a+2\,b+c \right) ^{-1}-{\frac {a+3\,c}{ \left( a+2\,b+c \right) ^{2}}}-8\,{\frac {b}{ \left( a+b+2\,c \right) ^{2}}}-8\, \left( a+b+3\,c \right) ^{-1}+24\,{\frac {c}{ \left( a+b+3\,c \right) ^{2}}}=0$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_53533 : ∀ a b c : ℝ, (a + 2 * b + c)⁻¹ - (a + 3 * c) / (a + 2 * b + c) ^ 2 - 4 * b / (a + b + 2 * c) ^ 2 + 8 * c / (a + b + 3 * c) ^ 2 = 0 ∧ -2 * (a + 3 * c) / (a + 2 * b + c) ^ 2 + 4 * (a + b + 2 * c)⁻¹ - 4 * b / (a + b + 2 * c) ^ 2 + 8 * c / (a + b + 3 * c) ^ 2 = 0 ∧ 3 * (a + 2 * b + c)⁻¹ - (a + 3 * c) / (a + 2 * b + c) ^ 2 - 8 * b / (a + b + 2 * c) ^ 2 - 8 * (a + b + 3 * c)⁻¹ + 24 * c / (a + b + 3 * c) ^ 2 = 0   :=  by sorry

Equations defining stationary points are asserted for every triple rather than posed as a system to solve.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

On the domain of nonzero denominators, solve the three displayed derivative equations, or prove that an interior critical point satisfies them.

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.