two_point_squared_entropy_le_half_sq_sub

Plain-English statement

Arbitrary signed a,b and zero cases; upper bound Ent(a squared,b squared) <= (a-b) squared/2. Minimal 26-helper positive-Rothaus port; upstream opposite-signed two_point_inequality unused. Independently verified 6d5df34cdb124e28022f16f566ff549145eb7e65. No GaussianLSI/T2/FIRST4.6/main/work/composition completion.

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.

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Lean statement

theorem two_point_squared_entropy_le_half_sq_sub (a b : ℝ) :
    (a ^ 2 * Real.log (a ^ 2) + b ^ 2 * Real.log (b ^ 2)) / 2 -
      ((a ^ 2 + b ^ 2) / 2) * Real.log ((a ^ 2 + b ^ 2) / 2) ≤
    (a - b) ^ 2 / 2 := by
  have hlog2 : Real.log 2 ≤ 1 := by
    have h := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2)
    norm_num at h
    exact h
  by_cases ha : a = 0
  · subst a
    by_cases hb : b = 0
    · simp [hb]
    · have hb2 : b ^ 2 ≠ 0 := pow_ne_zero 2 hb
      have hlog := Real.log_div hb2 (by norm_num : (2 : ℝ) ≠ 0)
      simp only [zero_pow (by norm_num : (2 : ℕ) ≠ 0), zero_mul, zero_add,
        zero_sub, neg_sq]
      rw [hlog]
      nlinarith [sq_nonneg b, mul_le_mul_of_nonneg_left hlog2 (sq_nonneg b)]
  · by_cases hb : b = 0
    · subst b
      have ha2 : a ^ 2 ≠ 0 := pow_ne_zero 2 ha
      have hlog := Real.log_div ha2 (by norm_num : (2 : ℝ) ≠ 0)
      simp only [zero_pow (by norm_num : (2 : ℕ) ≠ 0), zero_mul, add_zero, sub_zero]
      rw [hlog]
      nlinarith [sq_nonneg a, mul_le_mul_of_nonneg_left hlog2 (sq_nonneg a)]
    · have h := rothaus_lemma (sq_pos_of_ne_zero ha) (sq_pos_of_ne_zero hb)
      rw [Real.sqrt_sq_eq_abs, Real.sqrt_sq_eq_abs] at h
      have hab : (|a| - |b|)^2 ≤ (a-b)^2 := by
        rw [sq_le_sq]
        exact abs_abs_sub_abs_le_abs_sub a b
      linarith

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.TwoPointEntropy

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/TwoPointEntropy.lean:544published source at ed709a7f320d

Proof architecture

Actual SPHMC omitted Gaussian LSI background; no SALD or second-route admission

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