Tests.ProductEntropy

- The actual zero density exercises zero mass and every zero fiber.

theorem zero_product_entropy :
    let γ : Measure ℝ := gaussianReal 0 1
    let Φ : ℝ → ℝ := fun t => t * Real.log t
    Integrable (fun _ : ℝ × ℝ => (0 : ℝ)) (γ.prod γ) ∧
    (∫ _x : ℝ, Φ (∫ _y : ℝ, (0 : ℝ) ∂γ) ∂γ) +
      (∫ _y : ℝ, Φ (∫ _x : ℝ, (0 : ℝ) ∂γ) ∂γ) ≤
    (∫ _z : ℝ × ℝ, Φ 0 ∂γ.prod γ) + Φ (∫ _z : ℝ × ℝ, (0 : ℝ) ∂γ.prod γ) := by
  have h := bounded_product_entropy_subadditivity (gaussianReal 0 1) (gaussianReal 0 1)
    (fun _ : ℝ × ℝ => (0 : ℝ)) measurable_const (fun _ => le_refl 0) ⟨0, fun _ => le_refl 0⟩
  exact ⟨h.1, h.2.2.2.2.2.2.2.2⟩

Tests/ProductEntropy.lean:12published source at ed709a7f320d

No declaration docstring.

private def g (x : ℝ) : ℝ := x * smoothUnitCutoff x

Tests/ProductEntropy.lean:23published source at ed709a7f320d

No declaration docstring.

private def signedProbe (z : ℝ × ℝ) : ℝ := g z.1 * g z.2 + (g z.1 * g z.2)^2

Tests/ProductEntropy.lean:24published source at ed709a7f320d

No declaration docstring.

private def F (z : ℝ × ℝ) : ℝ := (signedProbe z)^2

/-- A concrete nonseparable squared signed compact observer. Its upper bound
and measurability are constructed here, without entropy/domain certificates. -/

Tests/ProductEntropy.lean:25published source at ed709a7f320d

- A concrete nonseparable squared signed compact observer. Its upper bound and measurability are constructed here, without entropy/domain certificates.

theorem actual_compact_product_entropy :
    let γ : Measure ℝ := gaussianReal 0 1
    let Φ : ℝ → ℝ := fun t => t * Real.log t
    (∫ x, Φ (∫ y, F (x, y) ∂γ) ∂γ) + (∫ y, Φ (∫ x, F (x, y) ∂γ) ∂γ) ≤
      (∫ z, Φ (F z) ∂γ.prod γ) + Φ (∫ z, F z ∂γ.prod γ) := by
  have cg : Continuous g := continuous_id.mul smoothUnitCutoff_contDiff.continuous
  have sg : HasCompactSupport g := smoothUnitCutoff_hasCompactSupport.mul_left
  obtain ⟨D, hD⟩ := sg.exists_bound_of_continuous cg
  let K := max D 0
  have hK : 0 ≤ K := le_max_right _ _
  have bg (x : ℝ) : ‖g x‖ ≤ K := (hD x).trans (le_max_left _ _)
  have bu (z : ℝ × ℝ) : ‖g z.1 * g z.2‖ ≤ K^2 := by
    rw [norm_mul, pow_two]
    exact mul_le_mul (bg z.1) (bg z.2) (norm_nonneg _) hK
  have bs (z : ℝ × ℝ) : ‖signedProbe z‖ ≤ K^2 + (K^2)^2 := by
    calc
      ‖signedProbe z‖ ≤ ‖g z.1 * g z.2‖ + ‖(g z.1 * g z.2)^2‖ := norm_add_le _ _
      _ ≤ K^2 + (K^2)^2 := by
        rw [norm_pow]
        exact add_le_add (bu z) (pow_le_pow_left₀ (norm_nonneg _) (bu z) 2)
  have cb : ∃ C : ℝ, ∀ z, F z ≤ C := by
    refine ⟨(K^2 + (K^2)^2)^2, ?_⟩
    intro z
    simpa only [F, Real.norm_eq_abs, sq_abs] using
      pow_le_pow_left₀ (norm_nonneg (signedProbe z)) (bs z) 2
  have mf : Measurable F := by
    apply Continuous.measurable
    exact ((cg.comp continuous_fst).mul (cg.comp continuous_snd) |>.add
      (((cg.comp continuous_fst).mul (cg.comp continuous_snd)).pow 2)).pow 2
  have h := bounded_product_entropy_subadditivity (gaussianReal 0 1) (gaussianReal 0 1)
    F mf (fun z => sq_nonneg (signedProbe z)) cb
  exact h.2.2.2.2.2.2.2.2

/-- A genuine zero fiber of the compact observer. -/

Tests/ProductEntropy.lean:29published source at ed709a7f320d

- A genuine zero fiber of the compact observer.

theorem compact_zero_fiber (y : ℝ) : F (0, y) = 0 := by simp [F, signedProbe, g]

/-- The observer really changes sign, while its square is nonnegative. -/

Tests/ProductEntropy.lean:63published source at ed709a7f320d

- The observer really changes sign, while its square is nonnegative.

theorem actual_probe_signed : 0 < signedProbe (1, 1) ∧ signedProbe (1/2, -1/2) < 0 := by
  norm_num [signedProbe, g,
    smoothUnitCutoff_eq_one_of_abs_le_one (x := 1) (by norm_num),
    smoothUnitCutoff_eq_one_of_abs_le_one (x := 1/2) (by norm_num),
    smoothUnitCutoff_eq_one_of_abs_le_one (x := -(1/2)) (by norm_num)]

/-- The consumer is an actual C2 compact observer on the product space. -/

Tests/ProductEntropy.lean:66published source at ed709a7f320d

- The consumer is an actual C2 compact observer on the product space.

theorem actual_probe_C2_compact : ContDiff ℝ 2 signedProbe ∧ HasCompactSupport signedProbe := by
  have sg : HasCompactSupport g := smoothUnitCutoff_hasCompactSupport.mul_left
  constructor
  · have cg : ContDiff ℝ 2 g := contDiff_id.mul
      (smoothUnitCutoff_contDiff.of_le (WithTop.coe_le_coe.mpr (le_top : (2 : ℕ∞) ≤ ⊤)))
    exact ((cg.comp contDiff_fst).mul (cg.comp contDiff_snd)).add
      (((cg.comp contDiff_fst).mul (cg.comp contDiff_snd)).pow 2)
  · apply HasCompactSupport.intro (sg.prod sg)
    intro z hz
    have ho : z.1 ∉ tsupport g ∨ z.2 ∉ tsupport g := by
      simpa only [Set.mem_prod, not_and_or] using hz
    rcases ho with hx | hy
    · simp [signedProbe, image_eq_zero_of_notMem_tsupport hx]
    · simp [signedProbe, image_eq_zero_of_notMem_tsupport hy]

#print axioms bounded_product_entropy_subadditivity
#print axioms zero_product_entropy
#print axioms actual_compact_product_entropy

end Tests.ProductEntropy

Tests/ProductEntropy.lean:73published source at ed709a7f320d