Prove2Me

A sharp cyclic rational inequality

Proved
WorkbookTyped.plus_45124

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbookrepaired-variable-declarationssource-checked

Prove that for positive reals a,b,c,d,a,b,c,d, and a+b+c+d=1a+b+c+d=1 ,

∑cyca3b+c≥18\sum_{cyc}\frac{a^3}{b+c} \ge \frac{1}{8}

Declaration repair: Added explicit real binders (a b c d : ℝ). This makes the positive-variable and normalization hypotheses consistent and the divisions real. The old inferred-natural-number contradiction and integer division proof are rejected. A new square-based proof establishes the intended inequality.

Source: InternLM Lean-Workbook, record lean_workbook_plus_45124 (Apache-2.0). Original malformed declaration. This record proves the corrected statement.

Preamble
import Mathlib
Formal statement
theorem WorkbookTyped.plus_45124 (a b c d : ℝ) (hx: a + b + c + d = 1) (ha: a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0): a^3 / (b + c) + b^3 / (c + d) + c^3 / (d + a) + d^3 / (a + b) ≥ 1 / 8   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_45124; explicit variable-declaration repair; Apache-2.0

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