`BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_achievable_iff` {c : ℕ} : (∃ p : ↥(stdSimplex ℝ (Fin n)), Nat.card ↥(bornMapSphere n ⁻¹' {p}) = c) ↔ ∃ k, 1 ≤ k ∧...

Proved

BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_achievable_iff

by leonardopedro · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornFiberSpectrum.

BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_achievable_iff {c : ℕ} : (∃ p : ↥(stdSimplex ℝ (Fin n)), Nat.card ↥(bornMapSphere n ⁻¹' {p}) = c) ↔ ∃ k, 1 ≤ k ∧ k ≤ n ∧ c = 2 ^ k

Formalization note: Lean 4 identifier BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_achievable_iff.

Preamble

-- Generated from ChapterFreeFieldBornFiberSpectrum.lean — theorem BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_achievable_iff
import Definitions.Def_ChapterFreeFieldBorn
import Definitions.Def_ChapterFreeFieldBornSurj
import Definitions.Def_ChapterFreeFieldBornCont
import Definitions.Def_ChapterFreeFieldBornQuotient
import Definitions.Def_ChapterFreeFieldBornFiberCardGeneral
import Definitions.Def_ChapterFreeFieldBornFiberTwo
import Definitions.Def_ChapterFreeFieldBornFiberBounds
import Mathlib
import Definitions.Def_ChapterFreeFieldBornFiberSpectrum
open BookProof.ChapterFreeFieldBornFiberSpectrum


open MeasureTheory
open BookProof.ChapterFreeFieldBorn BookProof.ChapterFreeFieldBornSurj
open BookProof.ChapterFreeFieldBornCont BookProof.ChapterFreeFieldBornQuotient
open BookProof.ChapterFreeFieldBornFiberCardGeneral
open BookProof.ChapterFreeFieldBornFiberTwo
open BookProof.ChapterFreeFieldBornFiberBounds


variable {n : ℕ}

Formal statement

theorem BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_achievable_iff {c : ℕ} :
    (∃ p : ↥(stdSimplex ℝ (Fin n)), Nat.card ↥(bornMapSphere n ⁻¹' {p}) = c) ↔
      ∃ k, 1 ≤ k ∧ k ≤ n ∧ c = 2 ^ k := by sorry

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