`BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_mono` {p q : ↥(stdSimplex ℝ (Fin n))} (h : posSupport (p : Fin n → ℝ) ⊆ posSupport (q : Fin n → ℝ)) : Nat.card ↥(bornMap

Proved

BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_mono

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornFiberSpectrum.

BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_mono {p q : ↥(stdSimplex ℝ (Fin n))} (h : posSupport (p : Fin n → ℝ) ⊆ posSupport (q : Fin n → ℝ)) : Nat.card ↥(bornMapSphere n ⁻¹' {p}) ≤ Nat.card ↥(bornMapSphere n ⁻¹' {q})

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

Preamble

-- Generated from ChapterFreeFieldBornFiberSpectrum.lean — theorem BookProof.ChapterFreeFieldBornFiberSpectrum.bornFiber_card_mono
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_mono {p q : ↥(stdSimplex ℝ (Fin n))}
    (h : posSupport (p : Fin n → ℝ) ⊆ posSupport (q : Fin n → ℝ)) :
    Nat.card ↥(bornMapSphere n ⁻¹' {p}) ≤ Nat.card ↥(bornMapSphere n ⁻¹' {q}) := by sorry

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