[{"type":"thinking","thinking":"Batch 1 compiles: `deriv2_of_hasDerivAt_nhds`, `psiOf`, `psiOf_id`, `psiOf_log_shift`. Also need sign lemmas. Let me now add the SIGN lemmas for id and log_shift, plus block L2 (psiOf_log_log_shift) with formula + sign. And think about consolidating: sign lemmas will share bounds (T ≥ √20+1 ⟹ T ≥ 5.4721, L1 ≥ 2, L2 ≥ 0.693). Prove a bounds lemma first:\n\n```lean\n/-- Bounds at the block-table range: T ≥ √20+1 ⟹ T ≥ 5.4721, L1 ≥ 2, L2 ≥ 0.6931. -/\ntheorem block_range_bounds {T : ℝ} (hT : Real.sqrt 20 + 1 ≤ T) :\n    5.4721 ≤ T ∧ 2 ≤ Real.log (T + 2) ∧ 0.6931 ≤ 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)]  -- ? signature: Real.le_sqrt (hy : 0 ≤ y) : x ≤ sqrt y ↔ x ^ 2 ≤ y? or (hx : 0 ≤ x)?\n    norm_num\n  ...\n```\nReal.le_sqrt signature: `Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ sqrt y ↔ x ^ 2 ≤ y`? or one side. There's `Real.le_sqrt' (hx : 0 < x) : x ≤ sqrt y ↔ x^2 ≤ y`? Let me just: have : (4.4721:ℝ)^2 ≤ 20 := by norm_num; exact Real.le_sqrt_of_sq_le?? `Real.le_sqrt_of_sq_le : x^2 ≤ y → x ≤ sqrt y`? Hmm `Real.le_sqrt_of_sq_le (h : x ^ 2 ≤ y) : x ≤ Real.sqrt y`? I believe `Real.le_sqrt_of_sq_le` exists (needs 0 ≤ x? maybe not since sqrt handles). Grep at write time; fallback: rw [Real.le_sqrt (by norm_num : (0:ℝ) ≤ 4.4721)]; norm_num — check Real.le_sqrt form.\n\nThen T ≥ 4.4721 + 1 = 5.4721 ✓ linarith.\nL1 = log(T+2) ≥ 2: T+2 ≥ 7.4721 ≥ e²: e² = (exp 1)² ≤ 2.7182818286² ≈ 7.389056...: exp_one_lt_d9 : Real.exp 1 < 2.7182818286 ✓. So (exp 1)^2 ≤ 2.7182818286^2 = 7.3890570... ≤ 7.4721 ≤ T+2. log(exp 2) = 2: exp 2 = (exp 1)^2: Real.exp_nat_mul? exp 2 = exp(1+1) = exp 1 * exp 1 ✓ Real.exp_add. log((exp 1)^2) = 2·log(exp 1) = 2·1 = 2 (Real.log_exp ✓). So:\nhave hL1 : (2:ℝ) ≤ Real.log (T + 2) := by\n  have h1 : Real.exp 2 ≤ T + 2 := by\n    have he : Real.exp 2 = Real.exp 1 * Real.exp 1 := by rw [← Real.exp_add]; norm_num\n    have hd : Real.exp 1 < 2.7182818286 := Real.exp_one_lt_d9\n    have hle : Real.exp 1 * Real.exp 1 ≤ 2.7182818286 * 2.7182818286 := by\n      apply mul_le_mul (le_of_lt hd) (le_of_lt hd) (by positivity) (by positivity... le_trans (Real.exp_pos 1).le hd.le) \n    ...\n  have h2 := Real.log_le_log (Real.exp_pos 2) h1  -- log (exp 2) ≤ log (T+2); name check\n  rwa [Real.log_exp] at h2\nReal.log_le_log signature: `Real.log_le_log (h₀ : 0 < x) (h₁ : x ≤ y) : log x ≤ log y`? or (0 < x) → (x ≤ y) → ...: there's `Real.log_le_log_iff (hy : 0 < y)`? and `Real.log_le_log_of_le`? Let me grep once. Also `Real.log_le_log (hx : 0 < x) (h : x ≤ y)` — pretty sure this exact form exists.\n\nL2 = log(L1) ≥ 0.6931: L1 ≥ 2 → log L1 ≥ log 2 ≥ 0.6931471803 > 0.6931: Real.log_two_gt_d9 : 0.6931471803 < Real.log 2 ✓. log_le_log (0 < 2) (L1 ≥ 2) ✓.\n\nThen the sign lemmas:\n\nsign_id: included in psiOf_id already (hT : 1 < T) ✓ done.\n\nsign_log_shift: for T ≥ √20+1 (or just 1 < T): −1/(T+2)² − 1/((T+2)(T−1)) < 0: both terms negative: 1/(T+2)² > 0 (sq pos), 1/((T+2)(T−1)) > 0 ✓. \n```lean\ntheorem psiOf_log_shift_neg {T : ℝ} (hT : 1 < T) :\n    psiOf (fun t => Real.log (t + 2)) T < 0 := by\n  rw [psiOf_log_shift T hT]\n  have h1 : (0:ℝ) < 1 / (T + 2)^2 := by positivity  -- needs T+2 ≠ 0: positivity uses... 1/(T+2)^2 > 0 always since (T+2)^2 > 0 (≠0): positivity might not know T+2 ≠ 0 — provide\n  ...\n```\npositivity for 1/(T+2)^2: (T+2)^2 ≥ 0, need > 0: T+2 ≠ 0 — positivity can't from context... use one_div_pos.mpr (pow_pos (by linarith) 2) ✓ explicit. Then linarith.\n\nBlock L2 (log log (t+2)): psiOf formula:\npsiOf (fun t => Real.log (Real.log (t+2))) T = -(L1+1)/((T+2)*L1)^2 - 1/((T+2)*L1*(T-1)) where L1 := Real.log (T+2).\n\nProof:\nh1 : ∀ᶠ x in 𝓝 T, HasDerivAt (fun t => Real.log (Real.log (t+2))) (((x+2) * Real.log (x+2))⁻¹) x:\n  filter_upwards [eventually_gt_nhds (-2 < T), eventually_gt_nhds (show (-1:ℝ) < T ...)] — need log(x+2) ≠ 0 ⟸ x+2 > 1 ⟸ x > −1: eventually x > −1 (T > 1 ≥ −1 ✓ use eventually_gt_nhds (by linarith : (-1:ℝ) < T)). Then per x: inner g1 = fun t => t+2 (HasDerivAt 1), g2 = log (point x+2 ≠ 0), g3 = log (point log(x+2) ≠ 0). \n  h_inner : HasDerivAt (fun t => Real.log (t+2)) (x+2)⁻¹ x := the comp thing (as before).\n  h_outer : HasDerivAt Real.log (Real.log (x+2))⁻¹ (Real.log (x+2)) := Real.hasDerivAt_log (log(x+2) ≠ 0 from x+2 > 1 → log > 0).\n  comp: HasDerivAt (log ∘ (fun t => log (t+2))) ((log(x+2))⁻¹ * (x+2)⁻¹) x — massage to ((x+2)*log(x+2))⁻¹: (log)⁻¹ * (x+2)⁻¹ = ((x+2)*log)⁻¹ by mul_inv_rev (comm: (a*b)⁻¹ = a⁻¹*b⁻¹: mul_inv : (a*b)⁻¹ = a⁻¹*b⁻¹ ✓): ((x+2)*log(x+2))⁻¹ = (x+2)⁻¹ * (log)⁻¹ — order: mul_inv gives (a*b)⁻¹ = a⁻¹*b⁻¹ — I have (log)⁻¹*(x+2)⁻¹ = ((log)*(x+2))⁻¹ reversed: rewrite goal form to match: state D1 x := (Real.log (x+2))⁻¹ * (x+2)⁻¹ instead of ((x+2)*log)⁻¹ — simpler! Let D1 x := (Real.log (x+2))⁻¹ * (x+2)⁻¹ and the final ψ-formula manipulated by field_simp at the end anyway.\n\nh2 : HasDerivAt D1 (−(L1+1)/((T+2)*L1)²) T where D1 t = (log(t+2))⁻¹ * (t+2)⁻¹:\n  D1 = product of u(t) = (log(t+2))⁻¹ and v(t) = (t+2)⁻¹.\n  u' = −(log(t+2))⁻²·(t+2)⁻¹ (from .inv of log-deriv: HasDerivAt (log(t+2)) (T+2)⁻¹ T → .inv (log(T+2)≠0) : HasDerivAt u (−(T+2)⁻¹/(log(T+2))²) T ✓).\n  v' = −1/(T+2)² (as before).\n  product rule: (u·v)' = u'·v + u·v' = [−(T+2)⁻¹/L1²]·(T+2)⁻¹ + L1⁻¹·(−(T+2)⁻²) = −1/((T+2)²L1²) − 1/((T+2)²L1) = −(1 + L1)/((T+2)²L1²) ✓.\n  Via HasDerivAt.mul hu hv : HasDerivAt (u*v) (u' * v T + u T * v') T — then massage with field_simp to −(L1+1)/((T+2)*L1)².\n\nThis is getting long — each block ~40-60 lines. There are 6 more blocks (L2, L1², L1L2, L2², TL1, TL2). ~300 lines total for formulas + signs. Proceed block by block. Write block L2 now.","signature":""},{"type":"text","text":"Batch 1 compiles. Adding sign lemmas + block L2 (formula + sign)."},{"type":"tool_use","id":"Edit_115","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":"/-- [PROVED session 011] Block #10 sign: ψ < 0 for T > 1. -/\ntheorem psiOf_log_shift_neg {T : ℝ} (hT : 1 < T) :\n    psiOf (fun t => Real.log (t + 2)) T < 0 := by\n  rw [psiOf_log_shift T hT]\n  have h1 : (0 : ℝ) < 1 / (T + 2) ^ 2 := one_div_pos.mpr (pow_pos (by linarith) 2)\n  have h2 : (0 : ℝ) < 1 / ((T + 2) * (T - 1)) :=\n    one_div_pos.mpr (mul_pos (by linarith) (by linarith))\n  linarith\n\n/-- [PROVED session 011] Block #11 (L2 = log log(T+2)): ψ-formula. -/\ntheorem psiOf_log_log_shift (T : ℝ) (hT : 1 < T) :\n    psiOf (fun t => Real.log (Real.log (t + 2))) T =\n      -(Real.log (T + 2) + 1) / ((T + 2) * Real.log (T + 2)) ^ 2 -\n        1 / ((T + 2) * Real.log (T + 2) * (T - 1)) := by\n  have hT2 : (0 : ℝ) < T + 2 := by linarith\n  have hL1 : (0 : ℝ) < Real.log (T + 2) := Real.log_pos (by linarith)\n  have h1 : ∀ᶠ x in 𝓝 T, HasDerivAt (fun t => Real.log (Real.log (t + 2)))\n      ((Real.log (x + 2))⁻¹ * (x + 2)⁻¹) x := by\n    filter_upwards [eventually_gt_nhds (by linarith : (-2 : ℝ) < T),\n      eventually_gt_nhds (by linarith : (-1 : ℝ) < T)] with x hx2 hx1\n    have hin : 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    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  have h2 : HasDerivAt (fun t => (Real.log (t + 2))⁻¹ * (t + 2)⁻¹)\n      (-(Real.log (T + 2) + 1) / ((T + 2) * Real.log (T + 2)) ^ 2) T := by\n    have hu : HasDerivAt (fun t => (Real.log (t + 2))⁻¹)\n        (-(T + 2)⁻¹ / Real.log (T + 2) ^ 2) T := by\n      have hin : HasDerivAt (fun t => Real.log (t + 2)) (T + 2)⁻¹ T := by\n        have h := (Real.hasDerivAt_log (show T + 2 ≠ 0 by linarith)).comp T\n          ((hasDerivAt_id' T).add_const 2)\n        rwa [mul_one] at h\n      have h := hin.inv (ne_of_gt hL1)\n      have e : -(T + 2)⁻¹ / Real.log (T + 2) ^ 2 =\n          -((T + 2)⁻¹ / Real.log (T + 2) ^ 2) := by ring\n      rwa [e] at h\n    have hv : 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 h := hu.mul hv\n    have hT2' : (T + 2) ≠ 0 := ne_of_gt hT2\n    have hL1' : Real.log (T + 2) ≠ 0 := ne_of_gt hL1\n    convert h using 2 <;> field_simp [hT2', hL1'] <;> ring\n  have hd2 := deriv2_of_hasDerivAt_nhds h1 h2\n  have hd1 : deriv (fun t => Real.log (Real.log (t + 2))) T =\n      (Real.log (T + 2))⁻¹ * (T + 2)⁻¹ := h1.self_of_nhds.deriv\n  unfold psiOf\n  rw [hd2, hd1]\n  have hT1 : (T - 1) ≠ 0 := by linarith\n  have hT2' : (T + 2) ≠ 0 := ne_of_gt hT2\n  have hL1' : Real.log (T + 2) ≠ 0 := ne_of_gt hL1\n  field_simp [hT1, hT2', hL1']\n  ring\n\n/-- [PROVED session 011] Block #11 sign: ψ < 0 for T > 1. -/\ntheorem psiOf_log_log_shift_neg {T : ℝ} (hT : 1 < T) :\n    psiOf (fun t => Real.log (Real.log (t + 2))) T < 0 := by\n  rw [psiOf_log_log_shift T hT]\n  have hL1 : (0 : ℝ) < Real.log (T + 2) := Real.log_pos (by linarith)\n  have hp1 : (0 : ℝ) < (Real.log (T + 2) + 1) / ((T + 2) * Real.log (T + 2)) ^ 2 :=\n    div_pos (by linarith) (pow_pos (mul_pos (by linarith) hL1) 2)\n  have hp2 : (0 : ℝ) < 1 / ((T + 2) * Real.log (T + 2) * (T - 1)) :=\n    one_div_pos.mpr (mul_pos (mul_pos (by linarith) hL1) (by linarith))\n  linarith\n\n/-- [SKELETON] Margin convexity (D-057, draft §3.1 Step 4): for any"},"extra_content":null}]