map_update_prod_pi · Samplinglib

Plain-English statement

- Replacing one coordinate of a product sample by an independent sample from that coordinate preserves the product law.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Lean learning studio · mathematics → formal proof

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Proof architecture

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Lean statement

theorem map_update_prod_pi (i : Fin n) :
    Measure.map (fun p : Ω × (Fin n → Ω) => Function.update p.2 i p.1)
      ((μs i).prod (Measure.pi μs)) = Measure.pi μs := by
  symm
  apply Measure.pi_eq
  intro s hs
  rw [Measure.map_apply]
  · have preimage_eq :
        (fun p : Ω × (Fin n → Ω) => Function.update p.2 i p.1) ⁻¹' (Set.univ.pi s) =
          (s i) ×ˢ (Set.univ.pi (fun j => if j = i then Set.univ else s j)) := by
      ext ⟨y, x⟩
      simp only [Set.mem_preimage, Set.mem_pi, Set.mem_univ, true_implies, Set.mem_prod]
      constructor
      · intro h
        constructor
        · have := h i
          simp only [Function.update_self] at this
          exact this
        · intro j
          by_cases hj : j = i
          · simp [hj]
          · have := h j
            simp only [Function.update_of_ne hj] at this
            simp [hj, this]
      · intro ⟨hy, hx⟩ j
        by_cases hj : j = i
        · subst hj
          simp only [Function.update_self]
          exact hy
        · simp only [Function.update_of_ne hj]
          have := hx j
          simp only [hj, ↓reduceIte] at this
          exact this
    rw [preimage_eq, Measure.prod_prod]
    have pi_eq :
        Measure.pi μs (Set.univ.pi (fun j => if j = i then Set.univ else s j)) =
          ∏ j : Fin n, (if j = i then 1 else μs j (s j)) := by
      rw [Measure.pi_pi]
      congr 1 with j
      by_cases hj : j = i
      · subst hj
        simp only [↓reduceIte, measure_univ]
      · simp only [hj, ↓reduceIte]
    rw [pi_eq]
    have h_ite : ∀ j, (if j = i then (1 : ℝ≥0∞) else μs j (s j)) =
        (if j ∈ Finset.univ.erase i then μs j (s j) else 1) := by
      intro j
      by_cases hj : j = i
      · simp [hj]
      · simp [hj]
    simp_rw [h_ite]
    rw [Fintype.prod_extend_by_one, mul_comm,
      Finset.prod_erase_mul _ _ (Finset.mem_univ i)]
  · exact measurable_update_prod_pi i
  · exact MeasurableSet.univ_pi (fun j => hs j)

/-- Measure-preserving wrapper for coordinate replacement under a product law. -/

Open AutoSamplingTheory/TechnicalLemmas/Measure/Product.lean:35published source at 7bcd37294df1

Proof architecture

Chewi MEAS/FI/SDE root: replace one coordinate of a finite product sample by an independent coordinate draw while preserving the product law

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`exact` closes the current goal with an already typed term.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.