`BookProof.ChapterEulerGenericDensity.conditional_probability_at` (θ : ℝ) (l w : Fin d → ℝ) (k : Fin d) (hlk : l k = 1) (hwk : w k = 0) : ((Real.cos θ ^ 2) • Matrix.vecMulVec l l +

Proved

BookProof.ChapterEulerGenericDensity.conditional_probability_at

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterEulerGenericDensity.

BookProof.ChapterEulerGenericDensity.conditional_probability_at (θ : ℝ) (l w : Fin d → ℝ) (k : Fin d) (hlk : l k = 1) (hwk : w k = 0) : ((Real.cos θ ^ 2) • Matrix.vecMulVec l l + (Real.sin θ ^ 2) • Matrix.vecMulVec w w) k k = Real.cos θ ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterEulerGenericDensity.conditional_probability_at.

Preamble

-- Generated from ChapterEulerGenericDensity.lean — theorem BookProof.ChapterEulerGenericDensity.conditional_probability_at
import Mathlib
import Definitions.Def_ChapterEulerGenericDensity
open BookProof.ChapterEulerGenericDensity


open scoped Matrix
open Matrix


variable {d : ℕ}

Formal statement

theorem BookProof.ChapterEulerGenericDensity.conditional_probability_at (θ : ℝ) (l w : Fin d → ℝ)
    (k : Fin d) (hlk : l k = 1) (hwk : w k = 0) :
    ((Real.cos θ ^ 2) • Matrix.vecMulVec l l
      + (Real.sin θ ^ 2) • Matrix.vecMulVec w w) k k = Real.cos θ ^ 2 := by sorry

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