[{"type":"thinking","thinking":"I left a sorry again in the middle of hInt2's final step — sloppy. Let me finish that step: (1/8)·x^(−σ−2) ≤ x^(−σ−2) — since 1/8 ≤ 1 and x^(−σ−2) ≥ 0: `mul_le_of_le_one_left (hb : 0 ≤ b) (h : a ≤ 1) : a * b ≤ b`? signature: mul_le_of_le_one_left : 0 ≤ b → a ≤ 1 → a * b ≤ b ✓. So:\n```lean\n        _ ≤ x ^ (-s.re - 2) := mul_le_of_le_one_left (by positivity) (by norm_num)\n```\nWait — positivity for 0 ≤ x^(−σ−2): rpow nonneg ✓ positivity handles (x > 0 in context? x > 1 from hx1 ✓ positivity can use? Real.rpow_nonneg needs 0 ≤ x: positivity uses hx1? Not automatically. Use `Real.rpow_nonneg (by linarith) _` explicitly.\n\nAlso the `simp` in the middle step: ‖(x:ℂ)^(−s−2)‖ = x^(re(−s−2)) = x^(−σ−2): norm_cpow_eq_rpow_re_of_pos (hx : 0 < x) : ‖(x:ℂ)^c‖ = x^(c.re) ✓; re(−s−2) = −σ−2: simp [sub_re, neg_re...] — the `simp` I wrote should handle re computation: (−s−2).re = −s.re − 2 ✓ simp lemmas sub_re/neg_re/ofNat_re... should work; maybe need explicit: rw [norm_cpow_eq_rpow_re_of_pos (by linarith)]; congr 1; simp? Let me restructure the calc to avoid fragility:\n\n```lean\n    · filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with x hx\n      have hx1 : (1:ℝ) < x := Set.mem_Ioi.mp hx\n      have hnorm : ‖(x : ℂ) ^ (-s - 2)‖ = x ^ (-s.re - 2) := by\n        rw [norm_cpow_eq_rpow_re_of_pos (by linarith)]\n        congr 2  -- re(−s−2) = −σ−2\n        simp  -- hmm: (−s−2).re = −(s.re) − 2: sub_re/neg_re\n      rw [norm_mul, Complex.norm_real, hnorm]\n      calc |b2Fun x| * x ^ (-s.re - 2)\n          ≤ (1 / 8) * x ^ (-s.re - 2) :=\n            mul_le_mul_of_nonneg_right (abs_b2Fun_le x) (Real.rpow_nonneg (by linarith) _)\n        _ ≤ x ^ (-s.re - 2) :=\n            mul_le_of_le_one_left (Real.rpow_nonneg (by linarith) _) (by norm_num)\n```\nAnd the target of mono': the bound function is x^(−σ−2) ✓.\n\nAlso `continuous_b2Fun.aestronglyMeasurable.ofReal` — Continuous.aestronglyMeasurable gives AEStronglyMeasurable b2Fun; .ofReal for AEStronglyMeasurable?? Earlier issue. Use: continuous of the ℂ function: (continuous_ofReal.comp continuous_b2Fun) — Continuous ⟹ aestronglyMeasurable: `(continuous_ofReal.comp continuous_b2Fun).aestronglyMeasurable` ✓ simpler. And the cpow piece: on Ioi 1: ContinuousOn: (continuousOn_of_continuousAt ...) — hmm is there `continuousOn_of_continuousAt`? Probably `ContinuousOn` from pointwise ContinuousWithinAt — actually `continuousOn_of_continuousAt`? hmm hmm. Let me use: `(continuousAt_ofReal_cpow_const ...)` per point → ContinuousWithinAt — then need ContinuousOn from ∀ x ∈ s, ContinuousWithinAt f s x — that's DEFINITIONALLY ContinuousOn ✓ (ContinuousOn f s := ∀ x ∈ s, ContinuousWithinAt f s x) ✓ so `fun x hx => (...).continuousWithinAt` IS a ContinuousOn proof ✓ (used in huu'-style above ✓).\n\nSo measurability: AEStronglyMeasurable (fun x => ↑b2Fun x * ↑x^(−s−2)) (volume.restrict (Ioi 1)): `AESTronglyMeasurable.mul`? Both aestronglyMeasurable on the restrict: hmm — IntegrableOn.mono' wants AEStronglyMeasurable f (μ.restrict s)... its signature: `IntegrableOn.mono' (h : IntegrableOn g s μ) (h1 : AEStronglyMeasurable f (μ.restrict s))?? (h2 : ∀ᵐ x ∂(μ.restrict s), ‖f x‖ ≤ g x)`? Let me check mono' signature — `IntegrableOn.mono' (h : IntegrableOn g s μ) (h1 : AEStronglyMeasurable f (μ.restrict s)) (h2 : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ g x)`? I believe: `IntegrableOn.mono' (h : IntegrableOn g s μ) (hf : AEStronglyMeasurable f (μ.restrict s)) (hfg : ∀ᵐ x ∂(μ.restrict s), ‖f x‖ ≤ g x) : IntegrableOn f s μ` ✓.\n\nSo the measurability arg: AEStronglyMeasurable (fun x => ↑b2Fun x * ↑x^(−s−2)) (volume.restrict (Ioi 1)): via `AESTronglyMeasurable.mul (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)`:\n- ↑(b2Fun): ((continuous_ofReal.comp continuous_b2Fun).aestronglyMeasurable) — that's on volume, need on restrict: `.mono_measure`? or AEStronglyMeasurable on restrict via `AESTronglyMeasurable.mono_measure (hm : μ' ≤ μ)`? restrict ≤ volume ✓ — there might be friction; alternative: prove AEStronglyMeasurable on the restrict directly from ContinuousOn: `ContinuousOn.aestronglyMeasurable (h : ContinuousOn f s) (hs : MeasurableSet s) : AEStronglyMeasurable f (μ.restrict s)` ✓ exists. b2Fun-ℂ is Continuous ⟹ ContinuousOn ✓. cpow: ContinuousOn via the per-point argument ✓.\n- then `.mul` ✓.\n\nOK let me write these carefully. Also the hInt1 part: `h.congr_fun ?_ measurableSet_Ioi` — IntegrableOn.congr_fun signature: `IntegrableOn.congr_fun (hf : IntegrableOn f s μ) (h : EqOn f g s) (hs : MeasurableSet s) : IntegrableOn g s μ` — direction: h : EqOn f g with hf about f proving g. h.neg : IntegrableOn (−their_integrand): want mine: EqOn (−their) (mine): ∀ x ∈ Ioi 1, −their x = mine x:\n```lean\n    refine h.congr_fun ?_ measurableSet_Ioi\n    intro x hx\n    -- goal: −((↑⌊x⌋ + 1/2 − ↑x) * ↑x^(−(s+1))) = ↑(B1 x) * ↑x^(−s−1)\n```\n`IntegrableOn.congr_fun (h : IntegrableOn f s μ) (hst : EqOn f g s) (hs : MeasurableSet s)` — hst : EqOn f g — so goal is `−(⋯) = ↑(B1 x) * ↑x ^ (-s - 1)` hmm — EqOn f g s unfolds to ∀ x ∈ s, f x = g x — intro x hx: goal f x = g x ✓.\n\nNow write the complete §2 ending. Also need `Real.rpow_nonneg`. Let me also double check `integrableOn_Ioi_rpow_of_lt {a c : ℝ} (ha : a < -1) (hc : 0 < c) : IntegrableOn (fun t => t ^ a) (Ioi c)` ✓ — call with a := -s.re - 2: need -s.re - 2 < -1 ⟺ s.re > -1 ✓ from hs : 0 < s.re by linarith ✓.","signature":""},{"type":"text","text":"Completing §2 properly (no placeholders this time)."},{"type":"tool_use","id":"Edit_234","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean","old_string":"  have hInt2 : IntegrableOn (fun x : ℝ => (b2Fun x : ℂ) * (x : ℂ) ^ (-s - 2)) (Set.Ioi 1) := by\n    apply IntegrableOn.mono' (integrableOn_Ioi_rpow_of_lt\n      (show -s.re - 2 < -1 by linarith) one_pos)\n    · exact (continuous_b2Fun.aestronglyMeasurable.ofReal.mul\n        ((continuousOn_of_continuousAt fun x hx =>\n          (continuousAt_ofReal_cpow_const x _ (Or.inr (ne_of_gt (Set.mem_Ioi.mp hx)))).continuousWithinAt\n          ).aestronglyMeasurable measurableSet_Ioi))\n    · filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with x hx\n      have hx1 : (1:ℝ) < x := Set.mem_Ioi.mp hx\n      rw [norm_mul, Complex.norm_real]\n      calc |b2Fun x| * ‖(x : ℂ) ^ (-s - 2)‖\n          ≤ (1 / 8) * ‖(x : ℂ) ^ (-s - 2)‖ :=\n            mul_le_mul_of_nonneg_right (abs_b2Fun_le x) (norm_nonneg _)\n        _ = (1 / 8) * x ^ (-s.re - 2) := by\n            rw [norm_cpow_eq_rpow_re_of_pos (by linarith)]\n            simp\n        _ ≤ x ^ (-s.re - 2) := by\n            apply mul_le_of_le_one_left (by positivity)\n            rw [Real.rpow_le_one_iff_one_le ... ] -- hmm\n            sorry\n  sorry","new_string":"  have hInt2 : IntegrableOn (fun x : ℝ => (b2Fun x : ℂ) * (x : ℂ) ^ (-s - 2)) (Set.Ioi 1) := by\n    apply IntegrableOn.mono' (integrableOn_Ioi_rpow_of_lt\n      (show -s.re - 2 < -1 by linarith) one_pos)\n    · exact ((continuous_ofReal.comp continuous_b2Fun).continuousOn.aestronglyMeasurable\n        measurableSet_Ioi).mul\n        ((fun x hx => (continuousAt_ofReal_cpow_const x _\n          (Or.inr (ne_of_gt (Set.mem_Ioi.mp hx)))).continuousWithinAt :\n            ContinuousOn (fun x : ℝ => (x : ℂ) ^ (-s - 2)) (Set.Ioi 1)).aestronglyMeasurable\n              measurableSet_Ioi)\n    · filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with x hx\n      have hx1 : (1:ℝ) < x := Set.mem_Ioi.mp hx\n      have hnorm : ‖(x : ℂ) ^ (-s - 2)‖ = x ^ (-s.re - 2) := by\n        rw [norm_cpow_eq_rpow_re_of_pos (by linarith)]\n        congr 2\n        simp\n      rw [norm_mul, Complex.norm_real, hnorm]\n      calc |b2Fun x| * x ^ (-s.re - 2)\n          ≤ (1 / 8) * x ^ (-s.re - 2) :=\n            mul_le_mul_of_nonneg_right (abs_b2Fun_le x) (Real.rpow_nonneg (by linarith) _)\n        _ ≤ x ^ (-s.re - 2) :=\n            mul_le_of_le_one_left (Real.rpow_nonneg (by linarith) _) (by norm_num)\n  have hT1 := intervalIntegral_tendsto_integral_Ioi (a := (1:ℝ)) (b := fun N : ℕ => (N : ℝ))\n    hInt1 tendsto_natCast_atTop_atTop\n  have hT2 := intervalIntegral_tendsto_integral_Ioi (a := (1:ℝ)) (b := fun N : ℕ => (N : ℝ))\n    hInt2 tendsto_natCast_atTop_atTop\n  have heq : ∀ᶠ N : ℕ in atTop,\n      (∫ x in (1:ℝ)..(N : ℝ), (B1 x : ℂ) * (x : ℂ) ^ (-s - 1))\n        = (s + 1) * ∫ x in (1:ℝ)..(N : ℝ), (b2Fun x : ℂ) * (x : ℂ) ^ (-s - 2) := by\n    filter_upwards [Filter.eventually_atTop.mpr ⟨1, fun N hN =>\n      intervalIntegral_b1_cpow_nat N hN s (fun h => by\n        have : s.re = -1 := by rw [h]; simp\n        linarith)⟩] with N hN\n    exact hN\n  exact tendsto_nhds_unique (hT1.congr' heq) (hT2.const_mul (s + 1))"},"extra_content":null}]