`BookProof.ChapterAttentionQKCircuit.headOutput_qkScore_congr` (beta : ℝ) {WQ₁ WK₁ WQ₂ WK₂ : Matrix (Fin d) (Fin n) ℝ} (h : qkMatrix WQ₁ WK₁ = qkMatrix WQ₂ WK₂) (x : Fin n...

Proved

BookProof.ChapterAttentionQKCircuit.headOutput_qkScore_congr

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionQKCircuit.

BookProof.ChapterAttentionQKCircuit.headOutput_qkScore_congr (beta : ℝ) {WQ₁ WK₁ WQ₂ WK₂ : Matrix (Fin d) (Fin n) ℝ} (h : qkMatrix WQ₁ WK₁ = qkMatrix WQ₂ WK₂) (x : Fin n → ℝ) (k : Fin m → Fin n → ℝ) (v : Fin m → E) : headOutput beta (fun l => qkScore WQ₁ WK₁ x (k l)) v = headOutput beta (fun l => qkScore WQ₂ WK₂ x (k l)) v

Formalization note: Lean 4 identifier BookProof.ChapterAttentionQKCircuit.headOutput_qkScore_congr.

Preamble

-- Generated from ChapterAttentionQKCircuit.lean — theorem BookProof.ChapterAttentionQKCircuit.headOutput_qkScore_congr
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionQKCircuit
import Definitions.Def_ChapterAttentionOutput
open BookProof.ChapterAttentionQKCircuit


open scoped BigOperators

noncomputable section


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

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

Formal statement

theorem BookProof.ChapterAttentionQKCircuit.headOutput_qkScore_congr (beta : ℝ) {WQ₁ WK₁ WQ₂ WK₂ : Matrix (Fin d) (Fin n) ℝ}
    (h : qkMatrix WQ₁ WK₁ = qkMatrix WQ₂ WK₂) (x : Fin n → ℝ) (k : Fin m → Fin n → ℝ)
    (v : Fin m → E) :
    headOutput beta (fun l => qkScore WQ₁ WK₁ x (k l)) v
      = headOutput beta (fun l => qkScore WQ₂ WK₂ x (k l)) v := by sorry

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