`BookProof.ChapterAttentionOVCircuit.mulVec_headOutput` (A : Matrix (Fin p) (Fin n) ℝ) (beta : ℝ) (s : Fin m → ℝ) (v : Fin m → (Fin n → ℝ)) : A *ᵥ headOutput beta s v = headOutput

Proved

BookProof.ChapterAttentionOVCircuit.mulVec_headOutput

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionOVCircuit.

BookProof.ChapterAttentionOVCircuit.mulVec_headOutput (A : Matrix (Fin p) (Fin n) ℝ) (beta : ℝ) (s : Fin m → ℝ) (v : Fin m → (Fin n → ℝ)) : A *ᵥ headOutput beta s v = headOutput beta s (fun j => A *ᵥ v j)

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

Preamble

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


open scoped BigOperators

noncomputable section


open Matrix BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterAttentionOutput

variable {d n p m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionOVCircuit.mulVec_headOutput (A : Matrix (Fin p) (Fin n) ℝ) (beta : ℝ) (s : Fin m → ℝ)
    (v : Fin m → (Fin n → ℝ)) :
    A *ᵥ headOutput beta s v = headOutput beta s (fun j => A *ᵥ v j) := by sorry

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