`BookProof.ChapterAttentionTopK.sum_le_sum_of_isTop` {p : Fin m → ℝ} (hp : ∀ x, 0 ≤ p x) {S T : Finset (Fin m)} (hS : IsTop p S) (hcard : T.card ≤ S.card) : ∑ x ∈ T, p x ≤ ∑ x ∈ S,

Proved

BookProof.ChapterAttentionTopK.sum_le_sum_of_isTop

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionTopK.

BookProof.ChapterAttentionTopK.sum_le_sum_of_isTop {p : Fin m → ℝ} (hp : ∀ x, 0 ≤ p x) {S T : Finset (Fin m)} (hS : IsTop p S) (hcard : T.card ≤ S.card) : ∑ x ∈ T, p x ≤ ∑ x ∈ S, p x

Formalization note: Lean 4 identifier BookProof.ChapterAttentionTopK.sum_le_sum_of_isTop.

Preamble

-- Generated from ChapterAttentionTopK.lean — theorem BookProof.ChapterAttentionTopK.sum_le_sum_of_isTop
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionTopK
open BookProof.ChapterAttentionTopK


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BookProof.ChapterAttentionTopK.sum_le_sum_of_isTop {p : Fin m → ℝ} (hp : ∀ x, 0 ≤ p x) {S T : Finset (Fin m)}
    (hS : IsTop p S) (hcard : T.card ≤ S.card) :
    ∑ x ∈ T, p x ≤ ∑ x ∈ S, p x := by sorry

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