`BookProof.ChapterCoherentOccupation.coherent_variance_lt_thermal_variance` {nbar : ℝ} (h : 0 < nbar) : ((∑' n : ℕ, (n : ℝ) ^ 2 * coherentOccupation nbar n) - (∑' n : ℕ, (n : ℝ) *

Proved

BookProof.ChapterCoherentOccupation.coherent_variance_lt_thermal_variance

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCoherentOccupation.

BookProof.ChapterCoherentOccupation.coherent_variance_lt_thermal_variance {nbar : ℝ} (h : 0 < nbar) : ((∑' n : ℕ, (n : ℝ) ^ 2 * coherentOccupation nbar n) - (∑' n : ℕ, (n : ℝ) * coherentOccupation nbar n) ^ 2) < ((∑' n : ℕ, (n : ℝ) ^ 2 * thermalProb nbar n) - (∑' n : ℕ, (n : ℝ) * thermalProb nbar n) ^ 2)

Formalization note: Lean 4 identifier BookProof.ChapterCoherentOccupation.coherent_variance_lt_thermal_variance.

Preamble

-- Generated from ChapterCoherentOccupation.lean — theorem BookProof.ChapterCoherentOccupation.coherent_variance_lt_thermal_variance
import Mathlib
import Definitions.Def_ChapterCoherentOccupation
import Definitions.Def_ChapterCoherentTemperature
open BookProof.ChapterCoherentTemperature
open BookProof.ChapterCoherentOccupation


noncomputable section


open Real Nat ProbabilityTheory

Formal statement

theorem BookProof.ChapterCoherentOccupation.coherent_variance_lt_thermal_variance {nbar : ℝ} (h : 0 < nbar) :
    ((∑' n : ℕ, (n : ℝ) ^ 2 * coherentOccupation nbar n)
        - (∑' n : ℕ, (n : ℝ) * coherentOccupation nbar n) ^ 2)
      < ((∑' n : ℕ, (n : ℝ) ^ 2 * thermalProb nbar n)
        - (∑' n : ℕ, (n : ℝ) * thermalProb nbar n) ^ 2) := by sorry

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