`BookProof.ChapterDisplacedThermalOverlap.dtBorn_temperature_coth` {x : ℝ} (hx : 0 < x) : ((BookProof.ChapterBoseEinstein.boseEinstein x).toNNReal : ℝ) + 1 / 2 = Real.cosh (x / 2)

Proved

BookProof.ChapterDisplacedThermalOverlap.dtBorn_temperature_coth

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDisplacedThermalOverlap.

BookProof.ChapterDisplacedThermalOverlap.dtBorn_temperature_coth {x : ℝ} (hx : 0 < x) : ((BookProof.ChapterBoseEinstein.boseEinstein x).toNNReal : ℝ) + 1 / 2 = Real.cosh (x / 2) / (2 * Real.sinh (x / 2))

Formalization note: Lean 4 identifier BookProof.ChapterDisplacedThermalOverlap.dtBorn_temperature_coth.

Preamble

-- Generated from ChapterDisplacedThermalOverlap.lean — theorem BookProof.ChapterDisplacedThermalOverlap.dtBorn_temperature_coth
import Mathlib
import Definitions.Def_ChapterDisplacedThermalOverlap
import Definitions.Def_ChapterBoseEinstein
open BookProof.ChapterBoseEinstein
open BookProof.ChapterDisplacedThermalOverlap


noncomputable section

open MeasureTheory ProbabilityTheory Real
open scoped NNReal

Formal statement

theorem BookProof.ChapterDisplacedThermalOverlap.dtBorn_temperature_coth {x : ℝ} (hx : 0 < x) :
    ((BookProof.ChapterBoseEinstein.boseEinstein x).toNNReal : ℝ) + 1 / 2
      = Real.cosh (x / 2) / (2 * Real.sinh (x / 2)) := by sorry

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