← Session 18 — Lean tier (ii): finite-height proof + certified base points
thinking (68 chars)
Now start the refactor. Edit 1: generalize chain_rule_sqrt_shift_on.
Starting the refactor. Edit 1: generalize `chain_rule_sqrt_shift_on` to any `c < 1`.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"old_string": "/-- [PROVED session 011] Localized variant of `chain_rule_sqrt_shift`:\n\u03c6 only needs to be C\u00b2 on (\u22121, \u221e) (all D-057 blocks: L1-singularities at\nT = \u22122, L2-singularities at T = \u22121). -/\ntheorem chain_rule_sqrt_shift_on (\u03c6 : \u211d \u2192 \u211d)\n (h\u03c6 : ContDiffOn \u211d 2 \u03c6 (Set.Ioi (-1))) (n : \u211d) (hn : 0 < n) :\n deriv (deriv fun n => \u03c6 (Real.sqrt n + 1)) n =\n (deriv (deriv \u03c6) (Real.sqrt n + 1) -\n deriv \u03c6 (Real.sqrt n + 1) / Real.sqrt n) / (4 * n) := by\n have h\u03c6d : DifferentiableOn \u211d \u03c6 (Set.Ioi (-1)) :=\n h\u03c6.differentiableOn (by norm_num)\n have h\u03c6dd : DifferentiableOn \u211d (deriv \u03c6) (Set.Ioi (-1)) :=\n (h\u03c6.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn\n one_ne_zero\n have hmem : \u2200 m : \u211d, 0 < m \u2192 Set.Ioi (-1) \u2208 \ud835\udcdd (Real.sqrt m + 1) := by\n intro m hm\n exact Ioi_mem_nhds (by linarith [Real.sqrt_nonneg m])",
"new_string": "/-- [PROVED session 011; generalized session 012 to any c < 1]\nLocalized variant of `chain_rule_sqrt_shift`: \u03c6 only needs to be C\u00b2 on\n(c, \u221e) with c < 1 (all D-057 blocks: L1-singularities at T = \u22122,\nL2-singularities at T = \u22121, so c = \u22121 suffices for them; the explicit\nTrudgian majorant FAlt of session 012 has log-singularities at 0 and 1,\nso it uses c = 1). -/\ntheorem chain_rule_sqrt_shift_on (\u03c6 : \u211d \u2192 \u211d) {c : \u211d} (hc : c < 1)\n (h\u03c6 : ContDiffOn \u211d 2 \u03c6 (Set.Ioi c)) (n : \u211d) (hn : 0 < n) :\n deriv (deriv fun n => \u03c6 (Real.sqrt n + 1)) n =\n (deriv (deriv \u03c6) (Real.sqrt n + 1) -\n deriv \u03c6 (Real.sqrt n + 1) / Real.sqrt n) / (4 * n) := by\n have h\u03c6d : DifferentiableOn \u211d \u03c6 (Set.Ioi c) :=\n h\u03c6.differentiableOn (by norm_num)\n have h\u03c6dd : DifferentiableOn \u211d (deriv \u03c6) (Set.Ioi c) :=\n (h\u03c6.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn\n one_ne_zero\n have hmem : \u2200 m : \u211d, 0 < m \u2192 Set.Ioi c \u2208 \ud835\udcdd (Real.sqrt m + 1) := by\n intro m hm\n exact Ioi_mem_nhds (by linarith [Real.sqrt_nonneg m])"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Now start the refactor. Edit 1: generalize chain_rule_sqrt_shift_on.","signature":""},{"type":"text","text":"Starting the refactor. Edit 1: generalize `chain_rule_sqrt_shift_on` to any `c < 1`."},{"type":"tool_use","id":"Edit_30","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- [PROVED session 011] Localized variant of `chain_rule_sqrt_shift`:\nφ only needs to be C² on (−1, ∞) (all D-057 blocks: L1-singularities at\nT = −2, L2-singularities at T = −1). -/\ntheorem chain_rule_sqrt_shift_on (φ : ℝ → ℝ)\n (hφ : ContDiffOn ℝ 2 φ (Set.Ioi (-1))) (n : ℝ) (hn : 0 < n) :\n deriv (deriv fun n => φ (Real.sqrt n + 1)) n =\n (deriv (deriv φ) (Real.sqrt n + 1) -\n deriv φ (Real.sqrt n + 1) / Real.sqrt n) / (4 * n) := by\n have hφd : DifferentiableOn ℝ φ (Set.Ioi (-1)) :=\n hφ.differentiableOn (by norm_num)\n have hφdd : DifferentiableOn ℝ (deriv φ) (Set.Ioi (-1)) :=\n (hφ.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn\n one_ne_zero\n have hmem : ∀ m : ℝ, 0 < m → Set.Ioi (-1) ∈ 𝓝 (Real.sqrt m + 1) := by\n intro m hm\n exact Ioi_mem_nhds (by linarith [Real.sqrt_nonneg m])","new_string":"/-- [PROVED session 011; generalized session 012 to any c < 1]\nLocalized variant of `chain_rule_sqrt_shift`: φ only needs to be C² on\n(c, ∞) with c < 1 (all D-057 blocks: L1-singularities at T = −2,\nL2-singularities at T = −1, so c = −1 suffices for them; the explicit\nTrudgian majorant FAlt of session 012 has log-singularities at 0 and 1,\nso it uses c = 1). -/\ntheorem chain_rule_sqrt_shift_on (φ : ℝ → ℝ) {c : ℝ} (hc : c < 1)\n (hφ : ContDiffOn ℝ 2 φ (Set.Ioi c)) (n : ℝ) (hn : 0 < n) :\n deriv (deriv fun n => φ (Real.sqrt n + 1)) n =\n (deriv (deriv φ) (Real.sqrt n + 1) -\n deriv φ (Real.sqrt n + 1) / Real.sqrt n) / (4 * n) := by\n have hφd : DifferentiableOn ℝ φ (Set.Ioi c) :=\n hφ.differentiableOn (by norm_num)\n have hφdd : DifferentiableOn ℝ (deriv φ) (Set.Ioi c) :=\n (hφ.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn\n one_ne_zero\n have hmem : ∀ m : ℝ, 0 < m → Set.Ioi c ∈ 𝓝 (Real.sqrt m + 1) := by\n intro m hm\n exact Ioi_mem_nhds (by linarith [Real.sqrt_nonneg m])"},"extra_content":null}]