lean_workbook_plus_45727

⚠️ 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.

Assume therefore $x,y,z\not =0$ and set $a={1\over x},b={1\over y}$ and $c={1\over z}$ .\nThen dividing 1),2) and 3) with $x^{2}y^{2}z^{2}$ gives\n4) $(b+c)^{2}=3+a+a^{2}$\n5) $(c+a)^{2}=4+b+b^{2}$\n6) $(a+b)^{2}=5+c+c^{2}$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_45727  (x y z a b c : ℝ)
  (h₀ : x ≠ 0)
  (h₁ : y ≠ 0)
  (h₂ : z ≠ 0)
  (h₃ : a = 1 / x)
  (h₄ : b = 1 / y)
  (h₅ : c = 1 / z)
  (h₆ : a * x = 1)
  (h₇ : b * y = 1)
  (h₈ : c * z = 1)
  (h₉ : a^2 + b^2 + c^2 = 1)
  (h₁₀ : a + b + c = 1)
  (h₁₁ : (b + c)^2 = 3 + a + a^2)
  (h₁₂ : (c + a)^2 = 4 + b + b^2)
  (h₁₃ : (a + b)^2 = 5 + c + c^2) :
  x = y ∧ y = z   :=  by sorry

The extra condition a+b+c=1 contradicts the three displayed transformed equations, so the equality conclusion is vacuous and unrelated to the excerpt’s reduction.

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

Recover equations(1)–(3) in x,y,z; with x,y,z≠0 and a=1/x,b=1/y,c=1/z, prove their equivalence to the three displayed transformed equations. Remove unsupported normalizations a+b+c=1 and a²+b²+c²=1 and do not assert x=y=z without the missing source.

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.