Adding three reciprocal relations

Proved

WorkbookSource.problem_45152

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

lean-workbooksource-checked

Put $a=xyz-{1\over xyz}$ . Then $x-{1\over y}={a\over 6}, y-{1\over z}={a\over 3}, z-{1\over x}={a\over 2}$ . Summing up those three we get $x+y+z-{1\over x}-{1\over y}-{1\over z}=a$

Source: InternLM Lean-Workbook, record lean_workbook_45152; Apache-2.0. Complete source proposition preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_45152  (x y z : ℝ)
  (a : ℝ)
  (h₀ : x ≠ 0 ∧ y ≠ 0 ∧ z ≠ 0)
  (h₁ : a = x * y * z - 1 / (x * y * z))
  (h₂ : x - 1/y = a/6)
  (h₃ : y - 1/z = a/3)
  (h₄ : z - 1/x = a/2) :
  x + y + z - 1/x - 1/y - 1/z = a  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_45152; Apache-2.0

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