`BookProof.ChapterCoherentThermalFidelity.dtOverlap_coherentParameter` (nb : NNReal) (a b : ℝ) : dtOverlap nb (Real.sqrt 2 * a) (Real.sqrt 2 * b) = Real.exp (-((a - b) ^ 2 / (((nb

Proved

BookProof.ChapterCoherentThermalFidelity.dtOverlap_coherentParameter

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCoherentThermalFidelity.

BookProof.ChapterCoherentThermalFidelity.dtOverlap_coherentParameter (nb : NNReal) (a b : ℝ) : dtOverlap nb (Real.sqrt 2 * a) (Real.sqrt 2 * b) = Real.exp (-((a - b) ^ 2 / (((nb : ℝ) + 1 / 2) + ((nb : ℝ) + 1 / 2)))) / Real.sqrt (4 * π * ((nb : ℝ) + 1 / 2))

Formalization note: Lean 4 identifier BookProof.ChapterCoherentThermalFidelity.dtOverlap_coherentParameter.

Preamble

-- Generated from ChapterCoherentThermalFidelity.lean — theorem BookProof.ChapterCoherentThermalFidelity.dtOverlap_coherentParameter
import Definitions.Def_ChapterCoherentOccupation
import Definitions.Def_ChapterCoherentTemperature
import Definitions.Def_ChapterCoherentFidelity
import Mathlib
import Definitions.Def_ChapterCoherentThermalFidelity
import Definitions.Def_ChapterDisplacedThermalOverlap
open BookProof.ChapterDisplacedThermalOverlap
open BookProof.ChapterCoherentThermalFidelity


noncomputable section


open BookProof.ChapterCoherentOccupation BookProof.ChapterCoherentTemperature
open BookProof.ChapterCoherentFidelity BookProof.ChapterDisplacedThermalOverlap
open Real

variable {nbar lam : ℝ}

Formal statement

theorem BookProof.ChapterCoherentThermalFidelity.dtOverlap_coherentParameter (nb : NNReal) (a b : ℝ) :
    dtOverlap nb (Real.sqrt 2 * a) (Real.sqrt 2 * b)
      = Real.exp (-((a - b) ^ 2 / (((nb : ℝ) + 1 / 2) + ((nb : ℝ) + 1 / 2))))
        / Real.sqrt (4 * π * ((nb : ℝ) + 1 / 2)) := by sorry

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