`BookProof.ChapterCoherentPositionSpace.integral_exp_mul_exp` (a b : ℝ) : (∫ x : ℝ, Real.exp (-(x - a) ^ 2 / 2) * Real.exp (-(x - b) ^ 2 / 2)) = Real.exp (-(a - b) ^ 2 / 4) * Real.

Proved

BookProof.ChapterCoherentPositionSpace.integral_exp_mul_exp

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCoherentPositionSpace.

BookProof.ChapterCoherentPositionSpace.integral_exp_mul_exp (a b : ℝ) : (∫ x : ℝ, Real.exp (-(x - a) ^ 2 / 2) * Real.exp (-(x - b) ^ 2 / 2)) = Real.exp (-(a - b) ^ 2 / 4) * Real.sqrt Real.pi

Formalization note: Lean 4 identifier BookProof.ChapterCoherentPositionSpace.integral_exp_mul_exp.

Preamble

-- Generated from ChapterCoherentPositionSpace.lean — theorem BookProof.ChapterCoherentPositionSpace.integral_exp_mul_exp
import Definitions.Def_ChapterCoherentOverlap
import Definitions.Def_ChapterSoftmaxSharpness
import Mathlib
import Definitions.Def_ChapterCoherentPositionSpace
open BookProof.ChapterCoherentPositionSpace


open scoped BigOperators
open MeasureTheory

noncomputable section


open BookProof.ChapterCoherentOverlap BookProof.ChapterSoftmaxSharpness

Formal statement

theorem BookProof.ChapterCoherentPositionSpace.integral_exp_mul_exp (a b : ℝ) :
    (∫ x : ℝ, Real.exp (-(x - a) ^ 2 / 2) * Real.exp (-(x - b) ^ 2 / 2))
      = Real.exp (-(a - b) ^ 2 / 4) * Real.sqrt Real.pi := by sorry

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