`BookProof.ChapterFreeFieldBornSectionBij.bornMap_bijOn_nonneg_sphere` : Set.BijOn (bornMap : EuclideanSpace ℝ (Fin n) → _) (Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1 ∩ nonneg

Proved

BookProof.ChapterFreeFieldBornSectionBij.bornMap_bijOn_nonneg_sphere

by leonardopedro · Oct 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornSectionBij.

BookProof.ChapterFreeFieldBornSectionBij.bornMap_bijOn_nonneg_sphere : Set.BijOn (bornMap : EuclideanSpace ℝ (Fin n) → _) (Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1 ∩ nonnegOrthant n) (stdSimplex ℝ (Fin n))

Formalization note: Lean 4 identifier BookProof.ChapterFreeFieldBornSectionBij.bornMap_bijOn_nonneg_sphere.

Preamble

-- Generated from ChapterFreeFieldBornSectionBij.lean — theorem BookProof.ChapterFreeFieldBornSectionBij.bornMap_bijOn_nonneg_sphere
import Definitions.Def_ChapterFreeFieldGaussian
import Definitions.Def_ChapterFreeFieldSphere
import Definitions.Def_ChapterFreeFieldSphereSupport
import Definitions.Def_ChapterFreeFieldBorn
import Definitions.Def_ChapterFreeFieldBornSurj
import Mathlib
import Definitions.Def_ChapterFreeFieldBornSectionBij
open BookProof.ChapterFreeFieldBornSectionBij

variable {n : ℕ}


open MeasureTheory
open BookProof.ChapterFreeFieldGaussian BookProof.ChapterFreeFieldSphere
open BookProof.ChapterFreeFieldSphereSupport BookProof.ChapterFreeFieldBorn
open BookProof.ChapterFreeFieldBornSurj

Formal statement

theorem BookProof.ChapterFreeFieldBornSectionBij.bornMap_bijOn_nonneg_sphere :
    Set.BijOn (bornMap : EuclideanSpace ℝ (Fin n) → _)
      (Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1 ∩ nonnegOrthant n)
      (stdSimplex ℝ (Fin n)) := by sorry

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