← Session 27 — ii′-5c/5d: Davenport disc + FE-reflection bounds
Now the `logDeriv_Gammaℝ_shift` rw restructure.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean",
"old_string": " have hdG4 : DifferentiableAt \u2102 (fun z : \u2102 => Complex.Gamma\u211d (z + 4)) s :=\n (differentiableAt_Gamma\u211d_of_im_ne_zero' (by simp [Complex.add_im]; exact hs)).comp s\n (differentiableAt_id.add_const 4)\n rw [logDeriv_congr hEv,\n logDeriv_div s (mul_ne_zero (pow_ne_zero 2 (by exact_mod_cast Real.pi_ne_zero)) hG4) hP\n ((differentiableAt_const _).mul hdG4)\n ((differentiableAt_id.div_const 2).mul (differentiableAt_id.div_const 2 |>.add_const 1)),\n logDeriv_const_mul s _ (pow_ne_zero 2 (by exact_mod_cast Real.pi_ne_zero : (Real.pi : \u2102) \u2260 0))]",
"new_string": " have hdG4 : DifferentiableAt \u2102 (fun z : \u2102 => Complex.Gamma\u211d (z + 4)) s :=\n (differentiableAt_Gamma\u211d_of_im_ne_zero' (by simp [Complex.add_im]; exact hs)).comp s\n (differentiableAt_id.add_const 4)\n have hpi_ne : (Real.pi : \u2102) \u2260 0 := by exact_mod_cast Real.pi_ne_zero\n have hnum_ne : (Real.pi : \u2102) ^ 2 * Complex.Gamma\u211d (s + 4) \u2260 0 :=\n mul_ne_zero (pow_ne_zero 2 hpi_ne) hG4\n have hnum_diff : DifferentiableAt \u2102\n (fun z => (Real.pi : \u2102) ^ 2 * Complex.Gamma\u211d (z + 4)) s :=\n (differentiableAt_const _).mul hdG4\n have hden_diff : DifferentiableAt \u2102 (fun z : \u2102 => (z / 2) * (z / 2 + 1)) s :=\n (differentiableAt_id.div_const 2).mul ((differentiableAt_id.div_const 2).add_const 1)\n have hstep1 : logDeriv (fun z => (Real.pi : \u2102) ^ 2 * Complex.Gamma\u211d (z + 4)\n / ((z / 2) * (z / 2 + 1))) s\n = logDeriv (fun z => (Real.pi : \u2102) ^ 2 * Complex.Gamma\u211d (z + 4)) s\n - logDeriv (fun z => (z / 2) * (z / 2 + 1)) s :=\n logDeriv_div s hnum_ne hP hnum_diff hden_diff\n have hstep2 : logDeriv (fun z => (Real.pi : \u2102) ^ 2 * Complex.Gamma\u211d (z + 4)) s\n = logDeriv (fun z => Complex.Gamma\u211d (z + 4)) s :=\n logDeriv_const_mul s _ (pow_ne_zero 2 hpi_ne)\n rw [logDeriv_congr hEv, hstep1, hstep2]"
}
raw content_json (database row)
[{"type":"text","text":"Now the `logDeriv_Gammaℝ_shift` rw restructure."},{"type":"tool_use","id":"Edit_56","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean","old_string":" have hdG4 : DifferentiableAt ℂ (fun z : ℂ => Complex.Gammaℝ (z + 4)) s :=\n (differentiableAt_Gammaℝ_of_im_ne_zero' (by simp [Complex.add_im]; exact hs)).comp s\n (differentiableAt_id.add_const 4)\n rw [logDeriv_congr hEv,\n logDeriv_div s (mul_ne_zero (pow_ne_zero 2 (by exact_mod_cast Real.pi_ne_zero)) hG4) hP\n ((differentiableAt_const _).mul hdG4)\n ((differentiableAt_id.div_const 2).mul (differentiableAt_id.div_const 2 |>.add_const 1)),\n logDeriv_const_mul s _ (pow_ne_zero 2 (by exact_mod_cast Real.pi_ne_zero : (Real.pi : ℂ) ≠ 0))]","new_string":" have hdG4 : DifferentiableAt ℂ (fun z : ℂ => Complex.Gammaℝ (z + 4)) s :=\n (differentiableAt_Gammaℝ_of_im_ne_zero' (by simp [Complex.add_im]; exact hs)).comp s\n (differentiableAt_id.add_const 4)\n have hpi_ne : (Real.pi : ℂ) ≠ 0 := by exact_mod_cast Real.pi_ne_zero\n have hnum_ne : (Real.pi : ℂ) ^ 2 * Complex.Gammaℝ (s + 4) ≠ 0 :=\n mul_ne_zero (pow_ne_zero 2 hpi_ne) hG4\n have hnum_diff : DifferentiableAt ℂ\n (fun z => (Real.pi : ℂ) ^ 2 * Complex.Gammaℝ (z + 4)) s :=\n (differentiableAt_const _).mul hdG4\n have hden_diff : DifferentiableAt ℂ (fun z : ℂ => (z / 2) * (z / 2 + 1)) s :=\n (differentiableAt_id.div_const 2).mul ((differentiableAt_id.div_const 2).add_const 1)\n have hstep1 : logDeriv (fun z => (Real.pi : ℂ) ^ 2 * Complex.Gammaℝ (z + 4)\n / ((z / 2) * (z / 2 + 1))) s\n = logDeriv (fun z => (Real.pi : ℂ) ^ 2 * Complex.Gammaℝ (z + 4)) s\n - logDeriv (fun z => (z / 2) * (z / 2 + 1)) s :=\n logDeriv_div s hnum_ne hP hnum_diff hden_diff\n have hstep2 : logDeriv (fun z => (Real.pi : ℂ) ^ 2 * Complex.Gammaℝ (z + 4)) s\n = logDeriv (fun z => Complex.Gammaℝ (z + 4)) s :=\n logDeriv_const_mul s _ (pow_ne_zero 2 hpi_ne)\n rw [logDeriv_congr hEv, hstep1, hstep2]"},"extra_content":null}]