`BookProof.ChapterDisplacedThermalOverlap.gaussianPDFReal_mul_gaussianPDFReal` (a b : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (x : ℝ) : gaussianPDFReal a v x * gaussianPDFReal b v x = (Real.exp

Proved

BookProof.ChapterDisplacedThermalOverlap.gaussianPDFReal_mul_gaussianPDFReal

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDisplacedThermalOverlap.

BookProof.ChapterDisplacedThermalOverlap.gaussianPDFReal_mul_gaussianPDFReal (a b : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (x : ℝ) : gaussianPDFReal a v x * gaussianPDFReal b v x = (Real.exp (-(a - b) ^ 2 / (4 * (v : ℝ))) / Real.sqrt (4 * π * v)) * gaussianPDFReal ((a + b) / 2) (v / 2) x

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

Preamble

-- Generated from ChapterDisplacedThermalOverlap.lean — theorem BookProof.ChapterDisplacedThermalOverlap.gaussianPDFReal_mul_gaussianPDFReal
import Mathlib
import Definitions.Def_ChapterDisplacedThermalOverlap
open BookProof.ChapterDisplacedThermalOverlap


noncomputable section

open MeasureTheory ProbabilityTheory Real
open scoped NNReal

Formal statement

theorem BookProof.ChapterDisplacedThermalOverlap.gaussianPDFReal_mul_gaussianPDFReal (a b : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (x : ℝ) :
    gaussianPDFReal a v x * gaussianPDFReal b v x
      = (Real.exp (-(a - b) ^ 2 / (4 * (v : ℝ))) / Real.sqrt (4 * π * v))
        * gaussianPDFReal ((a + b) / 2) (v / 2) x := by sorry

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