← Session 17 — Lean tier (i): elementary proofs (after crash)
thinking (185 chars)
`Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ √y ↔ x^2 ≤ y` — two args. Now write batch 2: the two standalone HasDerivAt helpers + block_range_bounds + L1², L1·L2, L2² formulas + signs.
Names confirmed. Writing batch 2: helpers + bounds lemma + blocks L1², L1·L2, L2² with signs.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"old_string": "/-- [PROVED session 011] Block #11 (L2 = log log(T+2)): \u03c8-formula. -/",
"new_string": "/-- [PROVED session 011] Reusable per-point derivative of log(t+2). -/\ntheorem hasDerivAt_log_shift (x : \u211d) (hx : -2 < x) :\n HasDerivAt (fun t => Real.log (t + 2)) (x + 2)\u207b\u00b9 x := by\n have h := (Real.hasDerivAt_log (show x + 2 \u2260 0 by linarith)).comp x\n ((hasDerivAt_id' x).add_const 2)\n rwa [mul_one] at h\n\n/-- [PROVED session 011] Reusable per-point derivative of log log(t+2). -/\ntheorem hasDerivAt_log_log (x : \u211d) (hx : -1 < x) :\n HasDerivAt (fun t => Real.log (Real.log (t + 2)))\n ((Real.log (x + 2))\u207b\u00b9 * (x + 2)\u207b\u00b9) x := by\n have hin := hasDerivAt_log_shift x (by linarith)\n have hout : HasDerivAt Real.log (Real.log (x + 2))\u207b\u00b9 (Real.log (x + 2)) :=\n Real.hasDerivAt_log (ne_of_gt (Real.log_pos (by linarith)))\n exact hout.comp x hin\n\n/-- [PROVED session 011] Numerical bounds on the block-table range\nT \u2265 \u221a20 + 1: T \u2265 5.4721, L1 \u2265 2 (via e\u00b2 \u2264 7.4721), L2 \u2265 0.6931. -/\ntheorem block_range_bounds {T : \u211d} (hT : Real.sqrt 20 + 1 \u2264 T) :\n (5.4721 : \u211d) \u2264 T \u2227 2 \u2264 Real.log (T + 2) \u2227\n (0.6931 : \u211d) \u2264 Real.log (Real.log (T + 2)) := by\n have hs : (4.4721 : \u211d) \u2264 Real.sqrt 20 := by\n rw [Real.le_sqrt (by norm_num) (by norm_num)]\n norm_num\n have hT5 : (5.4721 : \u211d) \u2264 T := by linarith\n have hT2 : (7.4721 : \u211d) \u2264 T + 2 := by linarith\n have hL1 : (2 : \u211d) \u2264 Real.log (T + 2) := by\n have he2 : Real.exp 2 \u2264 T + 2 := by\n have hd := Real.exp_one_lt_d9\n have e : Real.exp 2 = Real.exp 1 * Real.exp 1 := by\n rw [\u2190 Real.exp_add]; norm_num\n calc Real.exp 2 = Real.exp 1 * Real.exp 1 := e\n _ \u2264 2.7182818286 * 2.7182818286 :=\n mul_le_mul (le_of_lt hd) (le_of_lt hd) (by positivity) (Real.exp_pos 1).le\n _ \u2264 7.4721 := by norm_num\n _ \u2264 T + 2 := hT2\n have h2 := Real.log_le_log (Real.exp_pos 2) he2\n rwa [Real.log_exp] at h2\n have hL2 : (0.6931 : \u211d) \u2264 Real.log (Real.log (T + 2)) := by\n have h2 := Real.log_le_log (by norm_num : (0 : \u211d) < 2) hL1\n have h3 := Real.log_two_gt_d9\n linarith\n exact \u27e8hT5, hL1, hL2\u27e9\n\n/-- [PROVED session 011] Block #11 (L2 = log log(T+2)): \u03c8-formula. -/"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ √y ↔ x^2 ≤ y` — two args. Now write batch 2: the two standalone HasDerivAt helpers + block_range_bounds + L1², L1·L2, L2² formulas + signs.","signature":""},{"type":"text","text":"Names confirmed. Writing batch 2: helpers + bounds lemma + blocks L1², L1·L2, L2² with signs."},{"type":"tool_use","id":"Edit_132","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- [PROVED session 011] Block #11 (L2 = log log(T+2)): ψ-formula. -/","new_string":"/-- [PROVED session 011] Reusable per-point derivative of log(t+2). -/\ntheorem hasDerivAt_log_shift (x : ℝ) (hx : -2 < x) :\n HasDerivAt (fun t => Real.log (t + 2)) (x + 2)⁻¹ x := by\n have h := (Real.hasDerivAt_log (show x + 2 ≠ 0 by linarith)).comp x\n ((hasDerivAt_id' x).add_const 2)\n rwa [mul_one] at h\n\n/-- [PROVED session 011] Reusable per-point derivative of log log(t+2). -/\ntheorem hasDerivAt_log_log (x : ℝ) (hx : -1 < x) :\n HasDerivAt (fun t => Real.log (Real.log (t + 2)))\n ((Real.log (x + 2))⁻¹ * (x + 2)⁻¹) x := by\n have hin := hasDerivAt_log_shift x (by linarith)\n have hout : HasDerivAt Real.log (Real.log (x + 2))⁻¹ (Real.log (x + 2)) :=\n Real.hasDerivAt_log (ne_of_gt (Real.log_pos (by linarith)))\n exact hout.comp x hin\n\n/-- [PROVED session 011] Numerical bounds on the block-table range\nT ≥ √20 + 1: T ≥ 5.4721, L1 ≥ 2 (via e² ≤ 7.4721), L2 ≥ 0.6931. -/\ntheorem block_range_bounds {T : ℝ} (hT : Real.sqrt 20 + 1 ≤ T) :\n (5.4721 : ℝ) ≤ T ∧ 2 ≤ Real.log (T + 2) ∧\n (0.6931 : ℝ) ≤ Real.log (Real.log (T + 2)) := by\n have hs : (4.4721 : ℝ) ≤ Real.sqrt 20 := by\n rw [Real.le_sqrt (by norm_num) (by norm_num)]\n norm_num\n have hT5 : (5.4721 : ℝ) ≤ T := by linarith\n have hT2 : (7.4721 : ℝ) ≤ T + 2 := by linarith\n have hL1 : (2 : ℝ) ≤ Real.log (T + 2) := by\n have he2 : Real.exp 2 ≤ T + 2 := by\n have hd := Real.exp_one_lt_d9\n have e : Real.exp 2 = Real.exp 1 * Real.exp 1 := by\n rw [← Real.exp_add]; norm_num\n calc Real.exp 2 = Real.exp 1 * Real.exp 1 := e\n _ ≤ 2.7182818286 * 2.7182818286 :=\n mul_le_mul (le_of_lt hd) (le_of_lt hd) (by positivity) (Real.exp_pos 1).le\n _ ≤ 7.4721 := by norm_num\n _ ≤ T + 2 := hT2\n have h2 := Real.log_le_log (Real.exp_pos 2) he2\n rwa [Real.log_exp] at h2\n have hL2 : (0.6931 : ℝ) ≤ Real.log (Real.log (T + 2)) := by\n have h2 := Real.log_le_log (by norm_num : (0 : ℝ) < 2) hL1\n have h3 := Real.log_two_gt_d9\n linarith\n exact ⟨hT5, hL1, hL2⟩\n\n/-- [PROVED session 011] Block #11 (L2 = log log(T+2)): ψ-formula. -/"},"extra_content":null}]