`BookProof.ChapterAttentionOVCircuit.ovOutput_mem_range` (beta : ℝ) (s : Fin m → ℝ) (WO : Matrix (Fin n) (Fin d) ℝ) (WV : Matrix (Fin d) (Fin n) ℝ) (x : Fin m → (Fin n → ℝ)) : ovOu

Proved

BookProof.ChapterAttentionOVCircuit.ovOutput_mem_range

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionOVCircuit.

BookProof.ChapterAttentionOVCircuit.ovOutput_mem_range (beta : ℝ) (s : Fin m → ℝ) (WO : Matrix (Fin n) (Fin d) ℝ) (WV : Matrix (Fin d) (Fin n) ℝ) (x : Fin m → (Fin n → ℝ)) : ovOutput beta s WO WV x ∈ LinearMap.range (Matrix.mulVecLin WO)

Formalization note: Lean 4 identifier BookProof.ChapterAttentionOVCircuit.ovOutput_mem_range.

Preamble

-- Generated from ChapterAttentionOVCircuit.lean — theorem BookProof.ChapterAttentionOVCircuit.ovOutput_mem_range
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionOVCircuit
open BookProof.ChapterAttentionOVCircuit


open scoped BigOperators

noncomputable section


open Matrix BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {d n p m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionOVCircuit.ovOutput_mem_range (beta : ℝ) (s : Fin m → ℝ) (WO : Matrix (Fin n) (Fin d) ℝ)
    (WV : Matrix (Fin d) (Fin n) ℝ) (x : Fin m → (Fin n → ℝ)) :
    ovOutput beta s WO WV x ∈ LinearMap.range (Matrix.mulVecLin WO) := by sorry

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