`BookProof.EnergyBandDecomposition.mem_band_iff_floor` (hε : 0 < ε) (E : X → ℝ) (k : ℤ) (x : X) : x ∈ band E ε k ↔ k = ⌊E x / ε⌋

Proved

BookProof.EnergyBandDecomposition.mem_band_iff_floor

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterEnergyBandDecomposition.

BookProof.EnergyBandDecomposition.mem_band_iff_floor (hε : 0 < ε) (E : X → ℝ) (k : ℤ) (x : X) : x ∈ band E ε k ↔ k = ⌊E x / ε⌋

Formalization note: Lean 4 identifier BookProof.EnergyBandDecomposition.mem_band_iff_floor.

Preamble

-- Generated from ChapterEnergyBandDecomposition.lean — theorem BookProof.EnergyBandDecomposition.mem_band_iff_floor
import Mathlib
import Definitions.Def_ChapterEnergyBandDecomposition
open BookProof.EnergyBandDecomposition



open MeasureTheory

variable {X : Type*} {E : X → ℝ} {ε : ℝ} {k : ℤ} {x : X}

Formal statement

theorem BookProof.EnergyBandDecomposition.mem_band_iff_floor (hε : 0 < ε) (E : X → ℝ) (k : ℤ) (x : X) :
    x ∈ band E ε k ↔ k = ⌊E x / ε⌋ := by sorry

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