Prove2Me

`BookProof.ChapterMaxEntropy.negMulLog_sub_le` (n : ℕ) (hn : 0 < n) {x : ℝ} (hx : 0 ≤ x) : Real.negMulLog x - x * Real.log n ≤ (n : ℝ)⁻¹ - x

Proved
BookProof.ChapterMaxEntropy.negMulLog_sub_le

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMaxEntropy.

BookProof.ChapterMaxEntropy.negMulLog_sub_le (n : ℕ) (hn : 0 < n) {x : ℝ} (hx : 0 ≤ x) : Real.negMulLog x - x * Real.log n ≤ (n : ℝ)⁻¹ - x

Formalization note: Lean 4 identifier BookProof.ChapterMaxEntropy.negMulLog_sub_le.

Preamble
-- Generated from ChapterMaxEntropy.lean — theorem BookProof.ChapterMaxEntropy.negMulLog_sub_le
import Mathlib
import Definitions.Def_ChapterMaxEntropy
open BookProof.ChapterMaxEntropy


open Real BigOperators Finset


variable {α : Type*} [Fintype α]
Formal statement
theorem BookProof.ChapterMaxEntropy.negMulLog_sub_le (n : ℕ) (hn : 0 < n) {x : ℝ} (hx : 0 ≤ x) :
    Real.negMulLog x - x * Real.log n ≤ (n : ℝ)⁻¹ - x := by sorry

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