[{"type":"thinking","thinking":"`integrableOn_of_Zeta0_fun`: the integrand is `(⌊x⌋ + 1/2 - x) * (x:ℂ)^(-(s+1))` where `⌊x⌋` — with Int.measurable_floor used — so it's **Int.floor** there (⌊x⌋ with Int.measurable_floor)! Wait — the B1 def used ⌊x⌋₊ (Nat.floor) but this uses `Int.measurable_floor`... Let me look: `(⌊x⌋ + 1 / 2 - x)` — is that Nat or Int floor? The measurability proof: `Measurable.comp (by exact fun _ _ ↦ trivial) Int.measurable_floor` — so it's Int.floor here. But B1 uses ⌊x⌋₊ Nat.floor... So the Zeta0 integral uses Int-floor-based sawtooth (⌊x⌋ Int + 1/2 − x) while EulerMaclaurin's B1 uses Nat floor. For x ≥ 0 they agree. Our integrals are on Ioi 1 ⊆ positives ✓ fine either way — but for matching forms I need to know which floor is in the display. The display `(⌊x⌋ + 1 / 2 - x)` with Int.measurable_floor ⟹ Int.floor. So: their integrand = (↑⌊x⌋_Int + 1/2 − x)·x^(−(s+1)) = −(x − ⌊x⌋ − 1/2)·x^... = −(Int.fract x − 1/2)·x^... \n\nAnd B1 x = x − ↑⌊x⌋₊_Nat − 1/2 — for x ≥ 0 = Int.fract x − 1/2 (via natCast_floor_eq_intCast_floor ✓ that's exactly my b1_eq_fract_sub_half).\n\nSo for the hInt1 conversion: my integrand (B1 x)·x^(−s−1) vs theirs (⌊x⌋+1/2−x)·x^(−(s+1)): on Ioi 1: B1 x = fract x − 1/2 = −(⌊x⌋ + 1/2 − x) (Int version) ✓; exponents: −s−1 vs −(s+1) — equal by ring ✓. So:\n```lean\n  have hInt1 : IntegrableOn (fun x : ℝ => (B1 x : ℂ) * (x : ℂ) ^ (-s - 1)) (Set.Ioi 1) := by\n    have h := integrableOn_of_Zeta0_fun (N := 1) (by norm_num) (s := s) hs\n    -- h : IntegrableOn (fun x => (⌊x⌋ + 1/2 - x) * (x:ℂ)^(-(s+1))) (Ioi 1)\n    apply IntegrableOn.neg at h -- IntegrableOn (−their) \n    -- hmm: −((⌊x⌋+1/2−x)·x^...) = (x−⌊x⌋−1/2)·x^.. = (B1 x)·x^... on Ioi 1\n    refine h.congr' ?_  -- a.e. equality\n    -- hmm IntegrableOn.congr' : IntegrableOn f s μ → g =ᵐ[μ.restrict s] f → IntegrableOn g s μ?\n    ...\n```\n`IntegrableOn.congr'` direction: (hf : IntegrableOn f s μ) (h : f =ᵐ[μ.restrict s] g) : IntegrableOn g s μ? Let me use `IntegrableOn.congr_fun (hf) (h : ∀ x ∈ s, f x = g x) (measurable)`? There's `IntegrableOn.congr_fun (hf : IntegrableOn f s μ) (h : EqOn f g s) (hs : MeasurableSet s) : IntegrableOn g s μ` ✓ I think. So:\n```lean\n    apply (h.neg).congr_fun ?_ measurableSet_Ioi  -- direction: their-neg = mine\n    intro x hx\n    -- show: −((⌊x⌋+1/2−x)·x^(−(s+1))) = (B1 x)·x^(−s−1)\n    have hx0 : 0 ≤ x := (mem_Ioi.mp hx).le  -- x > 1\n    rw [b1_eq_fract_sub_half hx0, Int.fract]  -- B1 x = x − ⌊x⌋ − 1/2\n    push_cast\n    ring_nf  -- −((⌊x⌋+1/2−x)·X) = (x−⌊x⌋−1/2)·X with X = x^(−(s+1)) = x^(−s−1): need exponent match: −(s+1) = −s−1 by ring — ring_nf should handle the exponent? cpow exponent is inside an atom — ring_nf won't rewrite inside... Use congr on the cpow exponent first:\n```\nCleaner: first massage the exponent form: show (fun x => ...) equality with x^(−s−1) form matching via `show`+cong... Let me write it as: \n```lean\n    apply (h.neg).congr_fun ?_ measurableSet_Ioi\n    intro x hx\n    have hx0 : (0:ℝ) ≤ x := (Set.mem_Ioi.mp hx).le\n    simp only [Pi.neg_apply]\n    rw [b1_eq_fract_sub_half hx0, Int.fract]\n    have e : (-(s + 1) : ℂ) = -s - 1 := by ring\n    rw [e]  -- hmm rewrites all −(s+1) exponents... e : −(s+1) = −s−1: rw [e] rewrites LHS pattern — the exponent occurrence ✓\n    push_cast\n    ring\n```\nHmm — the cast structure: their integrand (⌊x⌋ + 1/2 − x) — the ⌊x⌋ is Int-floor cast? The display: `(⌊x⌋ + 1 / 2 - x) * (x : ℂ) ^ (-(s + 1))` — the ⌊x⌋ in a ℂ expression: ↑(⌊x⌋_Int)? Given Int.measurable_floor used, yes Int.floor. So the expression is (↑⌊x⌋ + 1/2 − ↑x)·↑x^(−(s+1)) with casts. My B1 x = x − ↑⌊x⌋₊_Nat − 1/2 (real), cast: ↑(B1 x) = (↑x − ↑⌊x⌋₊_Nat... — via b1_eq_fract_sub_half: B1 x = Int.fract x − 1/2 (REAL eq) — cast: ↑(B1 x) = ↑(fract x − 1/2) = ↑x − ↑⌊x⌋_Int − 1/2 (push_cast). So goal: −((↑⌊x⌋ + 1/2 − ↑x)·X) = (↑x − ↑⌊x⌋ − 1/2)·X with X = ↑x^(−s−1): ring ✓ (treat X atom — after exponent forms match via rw [e]).\n\nWait — the rw [e] approach: e : (-(s+1) : ℂ) = -s - 1 — rw [e] rewrites occurrences of −(s+1) in the GOAL — the goal at that point: −(↑⌊x⌋+1/2−↑x)·↑x^(−(s+1)) = ↑(B1 x)·↑x^(−s−1)... hmm I did rw [b1_eq...] turning ↑(B1 x) into ↑(fract−1/2)... then both sides have ↑x^(...) atoms with different exponent syntax: rw [e] turns −(s+1) into −s−1 ✓ then push_cast + ring with X atom ✓.\n\nThis is getting long — and I still have hInt2 + limit + §3 + §4 + §5. Realistic assessment: this file needs several more compile iterations. Keep going — write the rest of §2 (hInt2 + the limit theorem), compile, iterate. Let me replace the sorry block.","signature":""},{"type":"text","text":"Writing the complete limit theorem (replacing the sorry draft)."},{"type":"tool_use","id":"Edit_233","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean","old_string":"/-- **The level-2 Euler–Maclaurin identity** (re s > 0):\n∫₁^∞ B̃₁·x^{−s−1} = (s+1)·∫₁^∞ B̃₂·x^{−s−2}. -/\ntheorem integral_b1_cpow_eq {s : ℂ} (hs : 0 < s.re) :\n    ∫ x in Set.Ioi (1:ℝ), (B1 x : ℂ) * (x : ℂ) ^ (-s - 1)\n      = (s + 1) * ∫ x in Set.Ioi (1:ℝ), (b2Fun x : ℂ) * (x : ℂ) ^ (-s - 2) := by\n  have hInt1 : IntegrableOn (fun x : ℝ => (B1 x : ℂ) * (x : ℂ) ^ (-s - 1)) (Set.Ioi 1) := by\n    have h := integrableOn_of_Zeta0_fun (N := 1) (by norm_num) (s := s) hs\n    -- their integrand: (⌊x⌋ + 1/2 - x) * (x:ℂ)^(-(s+1)) = −B̃₁·x^{−s−1}\n    have heq : (fun x : ℝ => (B1 x : ℂ) * (x : ℂ) ^ (-s - 1))\n        = fun x : ℝ => -((⌊x⌋₊ + 1 / 2 - x : ℝ) : ℂ) * (x : ℂ) ^ (-(s + 1)) := by\n      funext x\n      rw [B1]\n      push_cast\n      ring_nf\n    rw [heq]\n    -- integrable from theirs via neg + exponent form\n    have h' : IntegrableOn (fun x : ℝ => -(((⌊x⌋₊ : ℝ) + 1 / 2 - x : ℝ) : ℂ) * (x : ℂ) ^ (-(s + 1)))\n        (Set.Ioi 1) := by\n      apply IntegrableOn.neg\n      convert h using 1\n      ext x\n      push_cast\n      ring_nf\n    exact h'\n  sorry\n  sorry","new_string":"/-- **The level-2 Euler–Maclaurin identity** (re s > 0):\n∫₁^∞ B̃₁·x^{−s−1} = (s+1)·∫₁^∞ B̃₂·x^{−s−2}. -/\ntheorem integral_b1_cpow_eq {s : ℂ} (hs : 0 < s.re) :\n    ∫ x in Set.Ioi (1:ℝ), (B1 x : ℂ) * (x : ℂ) ^ (-s - 1)\n      = (s + 1) * ∫ x in Set.Ioi (1:ℝ), (b2Fun x : ℂ) * (x : ℂ) ^ (-s - 2) := by\n  have hInt1 : IntegrableOn (fun x : ℝ => (B1 x : ℂ) * (x : ℂ) ^ (-s - 1)) (Set.Ioi 1) := by\n    have h := (integrableOn_of_Zeta0_fun (N := 1) (by norm_num) (s := s) hs).neg\n    refine h.congr_fun ?_ measurableSet_Ioi\n    intro x hx\n    have hx0 : (0:ℝ) ≤ x := (Set.mem_Ioi.mp hx).le\n    simp only [Pi.neg_apply]\n    rw [b1_eq_fract_sub_half hx0, Int.fract]\n    have e : (-(s + 1) : ℂ) = -s - 1 := by ring\n    rw [e]\n    push_cast\n    ring\n  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"},"extra_content":null}]