← Session 36 — ii′-6-DL dog-leg + side bounds; Paper A review
thinking (681 chars)
All signatures confirmed:
- `intervalIntegral.integral_eq_sub_of_hasDerivAt (hderiv : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a` ✓
- `HasDerivAt.sqrt (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun y => √(f y)) (f' / (2 * √(f x))) x` ✓
- `HasDerivAt.log (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun y => log (f y)) (f' / f x) x` ✓
- `intervalIntegral.integral_finsetSum (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b) : ∫ x in a..b, ∑ i ∈ s, f i x ∂μ = ∑ i ∈ s, ∫ x in a..b, f i x ∂μ` ✓
Now the DL-5 neg-side conj trick — let me read the hIntegral_Jlag_low_neg proof to clone the pattern.
All signatures confirmed. Now reading DL-5's neg-side conj trick.
tool: Read ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"offset": 1367,
"limit": 75
}
result 2026-08-26 18:13:22Z → 2026-08-26 18:13:22Z
{
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"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"content": "theorem hIntegral_Jlag_low_neg_norm_le_dogLegUpper {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 4 := by\n have hT0 : (0 : \u211d) < T := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsqrt4 : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n linarith\n have hconj : HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T) =\n (starRingEnd \u2102) (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T) := by\n unfold HIntegral\n rw [\u2190 intervalIntegral.intervalIntegral_conj]\n apply intervalIntegral.integral_congr\n intro \u03c3 h\u03c3\n show Jlag n ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n (starRingEnd \u2102) (Jlag n ((\u03c3 : \u211d) + T * Complex.I))\n have hid2 : ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n (starRingEnd \u2102) ((\u03c3 : \u211d) + T * Complex.I) := by\n simp\n rw [hid2]\n exact Jlag_conj n (by\n intro h\n have him : (((\u03c3 : \u211d) + T * Complex.I)).im = T := by simp\n rw [h] at him\n simp at him\n linarith [hT0])\n rw [hconj, Complex.norm_conj]\n exact hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1\n\n/-- **THE TWO-SIDE RIGHT-HORIZONTAL BOUND** (the DL capstone for\nii\u2032-6-c): both horizontal right tails bounded by the dog-leg charge. -/\ntheorem dogLegUpper_two_sides {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 +\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n 2 * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 4) := by\n rw [show (2 : \u211d) * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 +\n (8.38 : \u211d) + IIMaj (n : \u211d) / 4) =\n (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 4) +\n (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 4) by ring]\n exact add_le_add (hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1)\n (hIntegral_Jlag_low_neg_norm_le_dogLegUpper hn hTn hTn1)\n\n\nend NguyenRH1\n\nend",
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raw content_json (database row)
[{"type":"thinking","thinking":"All signatures confirmed:\n- `intervalIntegral.integral_eq_sub_of_hasDerivAt (hderiv : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a` ✓\n- `HasDerivAt.sqrt (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun y => √(f y)) (f' / (2 * √(f x))) x` ✓\n- `HasDerivAt.log (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun y => log (f y)) (f' / f x) x` ✓\n- `intervalIntegral.integral_finsetSum (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b) : ∫ x in a..b, ∑ i ∈ s, f i x ∂μ = ∑ i ∈ s, ∫ x in a..b, f i x ∂μ` ✓\n\nNow the DL-5 neg-side conj trick — let me read the hIntegral_Jlag_low_neg proof to clone the pattern.","signature":""},{"type":"text","text":"All signatures confirmed. Now reading DL-5's neg-side conj trick."},{"type":"tool_use","id":"Read_91","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","offset":1367,"limit":75},"extra_content":null}]