← Session 17 — Lean tier (i): elementary proofs (after crash)
thinking (110 chars)
Batch 1: helper `deriv2_of_hasDerivAt_nhds` + `psiOf` def + blocks id, L1, L2 (ψ formulas + signs). Write now.
Batch 1 of the block table: helper + `psiOf` + blocks id/L1/L2.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"old_string": "/-- [SKELETON] Margin convexity (D-057, draft \u00a73.1 Step 4): for any",
"new_string": "/-! ### Block machinery for the `marginTilde_convex` proof (D-057 \u00a72) -/\n\n/-- Second-derivative computation package: if \u03c6 has derivative D1 on a\nneighborhood of T and D1 has derivative D2 at T, then \u03c6\u2033(T) = D2. -/\ntheorem deriv2_of_hasDerivAt_nhds {\u03c6 D1 : \u211d \u2192 \u211d} {T D2 : \u211d}\n (h1 : \u2200\u1da0 x in \ud835\udcdd T, HasDerivAt \u03c6 (D1 x) x) (h2 : HasDerivAt D1 D2 T) :\n deriv (deriv \u03c6) T = D2 :=\n (h2.congr_of_eventuallyEq (by\n filter_upwards [h1] with x hx using hx.deriv.symm)).deriv\n\n/-- The \u03c8-functional of D-057: \u03c8[\u03c6](T) = \u03c6\u2033(T) \u2212 \u03c6\u2032(T)/(T\u22121). -/\ndef psiOf (\u03c6 : \u211d \u2192 \u211d) (T : \u211d) : \u211d := deriv (deriv \u03c6) T - deriv \u03c6 T / (T - 1)\n\n/-- [PROVED session 011] Block #2 (identity): \u03c8[id](T) = \u22121/(T\u22121) < 0. -/\ntheorem psiOf_id (T : \u211d) (hT : 1 < T) : psiOf id T = -1 / (T - 1) \u2227 psiOf id T < 0 := by\n have h1 : deriv (deriv id) T = 0 := by\n have he : deriv id =\u1da0[\ud835\udcdd T] fun _ => (1 : \u211d) :=\n eventually_of_forall fun _ => deriv_id'' _\n rw [he.deriv, deriv_const]\n have h2 : deriv id T = 1 := deriv_id'' T\n have h3 : (0 : \u211d) < T - 1 := by linarith\n constructor\n \u00b7 unfold psiOf; rw [h1, h2]\n \u00b7 unfold psiOf; rw [h1, h2]; exact neg_neg_of_pos (one_div_pos.mpr h3)\n\n/-- [PROVED session 011] Block #10 (L1 = log(T+2)): \u03c8-formula and sign. -/\ntheorem psiOf_log_shift (T : \u211d) (hT : -1 < T) :\n psiOf (fun t => Real.log (t + 2)) T =\n -1 / (T + 2) ^ 2 - 1 / ((T + 2) * (T - 1)) := by\n have hT2 : (0 : \u211d) < T + 2 := by linarith\n have h1 : \u2200\u1da0 x in \ud835\udcdd T, HasDerivAt (fun t => Real.log (t + 2)) ((x + 2)\u207b\u00b9) x := by\n filter_upwards [eventually_gt_nhds hT2] with x hx\n exact (Real.hasDerivAt_log (by linarith)).add_const 2\n have h2 : HasDerivAt (fun t => (t + 2)\u207b\u00b9) (-1 / (T + 2) ^ 2) T := by\n have h := ((hasDerivAt_id' T).add_const 2).inv (by linarith : (T + 2) \u2260 0)\n have e : (-(1 : \u211d)) / (T + 2) ^ 2 = -1 / (T + 2) ^ 2 := by ring\n rwa [e] at h\n have hd2 := deriv2_of_hasDerivAt_nhds h1 h2\n have hd1 : deriv (fun t => Real.log (t + 2)) T = (T + 2)\u207b\u00b9 :=\n (h1.self_of_nhdsWithin??).deriv\n sorry\n\n/-- [SKELETON] Margin convexity (D-057, draft \u00a73.1 Step 4): for any"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Batch 1: helper `deriv2_of_hasDerivAt_nhds` + `psiOf` def + blocks id, L1, L2 (ψ formulas + signs). Write now.","signature":""},{"type":"text","text":"Batch 1 of the block table: helper + `psiOf` + blocks id/L1/L2."},{"type":"tool_use","id":"Edit_98","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- [SKELETON] Margin convexity (D-057, draft §3.1 Step 4): for any","new_string":"/-! ### Block machinery for the `marginTilde_convex` proof (D-057 §2) -/\n\n/-- Second-derivative computation package: if φ has derivative D1 on a\nneighborhood of T and D1 has derivative D2 at T, then φ″(T) = D2. -/\ntheorem deriv2_of_hasDerivAt_nhds {φ D1 : ℝ → ℝ} {T D2 : ℝ}\n (h1 : ∀ᶠ x in 𝓝 T, HasDerivAt φ (D1 x) x) (h2 : HasDerivAt D1 D2 T) :\n deriv (deriv φ) T = D2 :=\n (h2.congr_of_eventuallyEq (by\n filter_upwards [h1] with x hx using hx.deriv.symm)).deriv\n\n/-- The ψ-functional of D-057: ψ[φ](T) = φ″(T) − φ′(T)/(T−1). -/\ndef psiOf (φ : ℝ → ℝ) (T : ℝ) : ℝ := deriv (deriv φ) T - deriv φ T / (T - 1)\n\n/-- [PROVED session 011] Block #2 (identity): ψ[id](T) = −1/(T−1) < 0. -/\ntheorem psiOf_id (T : ℝ) (hT : 1 < T) : psiOf id T = -1 / (T - 1) ∧ psiOf id T < 0 := by\n have h1 : deriv (deriv id) T = 0 := by\n have he : deriv id =ᶠ[𝓝 T] fun _ => (1 : ℝ) :=\n eventually_of_forall fun _ => deriv_id'' _\n rw [he.deriv, deriv_const]\n have h2 : deriv id T = 1 := deriv_id'' T\n have h3 : (0 : ℝ) < T - 1 := by linarith\n constructor\n · unfold psiOf; rw [h1, h2]\n · unfold psiOf; rw [h1, h2]; exact neg_neg_of_pos (one_div_pos.mpr h3)\n\n/-- [PROVED session 011] Block #10 (L1 = log(T+2)): ψ-formula and sign. -/\ntheorem psiOf_log_shift (T : ℝ) (hT : -1 < T) :\n psiOf (fun t => Real.log (t + 2)) T =\n -1 / (T + 2) ^ 2 - 1 / ((T + 2) * (T - 1)) := by\n have hT2 : (0 : ℝ) < T + 2 := by linarith\n have h1 : ∀ᶠ x in 𝓝 T, HasDerivAt (fun t => Real.log (t + 2)) ((x + 2)⁻¹) x := by\n filter_upwards [eventually_gt_nhds hT2] with x hx\n exact (Real.hasDerivAt_log (by linarith)).add_const 2\n have h2 : HasDerivAt (fun t => (t + 2)⁻¹) (-1 / (T + 2) ^ 2) T := by\n have h := ((hasDerivAt_id' T).add_const 2).inv (by linarith : (T + 2) ≠ 0)\n have e : (-(1 : ℝ)) / (T + 2) ^ 2 = -1 / (T + 2) ^ 2 := by ring\n rwa [e] at h\n have hd2 := deriv2_of_hasDerivAt_nhds h1 h2\n have hd1 : deriv (fun t => Real.log (t + 2)) T = (T + 2)⁻¹ :=\n (h1.self_of_nhdsWithin??).deriv\n sorry\n\n/-- [SKELETON] Margin convexity (D-057, draft §3.1 Step 4): for any"},"extra_content":null}]