`BookProof.ChapterF6.mgStep_le_apply_add` (k : ℕ) (T : α →₀ ℕ) (x y : α) : T y + (if y = x then 1 else 0) ≤ (mgStep k T x) y + (if 0 < T x ∨ T.support.card < k then 0 else 1)

Proved

BookProof.ChapterF6.mgStep_le_apply_add

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterF6.

BookProof.ChapterF6.mgStep_le_apply_add (k : ℕ) (T : α →₀ ℕ) (x y : α) : T y + (if y = x then 1 else 0) ≤ (mgStep k T x) y + (if 0 < T x ∨ T.support.card < k then 0 else 1)

Formalization note: Lean 4 identifier BookProof.ChapterF6.mgStep_le_apply_add.

Preamble

-- Generated from ChapterF6.lean — theorem BookProof.ChapterF6.mgStep_le_apply_add
import Mathlib
import Definitions.Def_ChapterF6
open BookProof.ChapterF6


open scoped BigOperators


variable {α : Type*} [DecidableEq α]

Formal statement

theorem BookProof.ChapterF6.mgStep_le_apply_add (k : ℕ) (T : α →₀ ℕ) (x y : α) :
    T y + (if y = x then 1 else 0)
      ≤ (mgStep k T x) y + (if 0 < T x ∨ T.support.card < k then 0 else 1) := by sorry

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