`BookProof.ChapterCoherentGeometry.coherentOverlap_le_iff_dist_le` (q k k' : EuclideanSpace ℝ (Fin n)) : coherentOverlap q k ≤ coherentOverlap q k' ↔ ‖q - k'‖ ≤ ‖q - k‖

Proved

BookProof.ChapterCoherentGeometry.coherentOverlap_le_iff_dist_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCoherentGeometry.

BookProof.ChapterCoherentGeometry.coherentOverlap_le_iff_dist_le (q k k' : EuclideanSpace ℝ (Fin n)) : coherentOverlap q k ≤ coherentOverlap q k' ↔ ‖q - k'‖ ≤ ‖q - k‖

Formalization note: Lean 4 identifier BookProof.ChapterCoherentGeometry.coherentOverlap_le_iff_dist_le.

Preamble

-- Generated from ChapterCoherentGeometry.lean — theorem BookProof.ChapterCoherentGeometry.coherentOverlap_le_iff_dist_le
import Definitions.Def_ChapterSoftmaxBorn
import Mathlib
import Definitions.Def_ChapterCoherentGeometry
import Definitions.Def_ChapterCoherentOverlap
open BookProof.ChapterCoherentOverlap
open BookProof.ChapterCoherentGeometry


open scoped BigOperators

noncomputable section


open BookProof.ChapterCoherentOverlap BookProof.ChapterSoftmaxBorn

variable {n m : ℕ}

Formal statement

theorem BookProof.ChapterCoherentGeometry.coherentOverlap_le_iff_dist_le (q k k' : EuclideanSpace ℝ (Fin n)) :
    coherentOverlap q k ≤ coherentOverlap q k' ↔ ‖q - k'‖ ≤ ‖q - k‖ := by sorry

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