[{"type":"thinking","thinking":"Wait — in Bn_FAlt_100000_le, the hlogTge1/hlogT2ge1 proofs have a syntax error: `(by have h := Real.sqrt_nonneg 100000; norm_num : (2.7182818286 : ℝ) ≤ 317)` — that's malformed (a by-block with a type ascription after — invalid syntax: `by ... : type` isn't valid there). The intent: prove exp 1 ≤ √100000+1 via le_trans exp_one_lt_d9.le (h : 2.7182818286 ≤ √100000+1). The h : 2.7182818286 ≤ √100000+1 needs √100000 ≥ 2.7182 − 1 = 1.7182 — TRUE trivially (√100000 ≥ 0 is too weak; √100000 ≥ 316 — but I only proved sqrt100000_le (≤ 317). Need a lower bound: √100000 ≥ 316 ⟸ 316² = 99856 ≤ 100000 ✓. I never proved sqrt100000_ge! Add it:\n```\ntheorem sqrt100000_ge : (316 : ℝ) ≤ Real.sqrt 100000 :=\n  (Real.le_sqrt (by norm_num) (by norm_num)).mpr (by norm_num)\n```\nThen hlogTge1: exp 1 ≤ √100000+1 ⟸ 2.7182818286 ≤ 317 ≤ √100000+1: le_trans Real.exp_one_lt_d9.le (by linarith [sqrt100000_ge]).\n\nSimilarly for the 80000 case: hlogTge1 used le_trans Real.exp_one_lt_d9.le (by linarith [sqrt80000_ge]) — 2.7182818286 ≤ 283 ≤ √80000+1 ✓ that's what I wrote for the 80000 case ✓ (that one is fine).\n\nFix the two malformed haves in the 100000 case and add sqrt100000_ge. Also check hloglogT's Real.log_pos (by linarith [Real.sqrt_nonneg 100000]) — goal 1 < √100000+1 ⟸ 0 < √100000: sqrt_nonneg gives ≥ 0 only — INSUFFICIENT (need strict > 0)! √100000 > 0 from sqrt100000_ge (316 ≤ √100000) → linarith [sqrt100000_ge]. Fix: Real.log_pos (by linarith [sqrt100000_ge]).\n\nSame in the 80000 case: hlogT2pos used Real.log_pos (by linarith [sqrt80000_ge]) ✓ (fixed earlier). And hlogT2nn : log_nonneg (1 ≤ √+1+2): 1 ≤ √80000+1+2 ⟸ 0 ≤ √80000+1 — from hTpos? In the 80000_ge proof hlogTnn used `Real.log_nonneg (by linarith)` with hTge : 283 ≤ √80000+1 in context ✓ fine. In the 80000_le proof hlogTnn : Real.log_nonneg (by linarith [Real.sqrt_nonneg 80000]) — goal 1 ≤ √80000+1 ⟺ 0 ≤ √80000 ✓ sqrt_nonneg ✓ OK.\n\nLet me fix the two malformed blocks.","signature":""},{"type":"text","text":"Two malformed blocks in the 100000-case (stray type ascriptions in by-blocks) plus a missing `sqrt100000_ge`. Fixing."},{"type":"tool_use","id":"Edit_164","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean","old_string":"/-- [PROVED session 012] √100000 ≤ 317 (317² = 100489). -/\ntheorem sqrt100000_le : Real.sqrt 100000 ≤ 317 :=\n  (Real.sqrt_le_left (by norm_num)).mpr (by norm_num)","new_string":"/-- [PROVED session 012] √100000 ≤ 317 (317² = 100489). -/\ntheorem sqrt100000_le : Real.sqrt 100000 ≤ 317 :=\n  (Real.sqrt_le_left (by norm_num)).mpr (by norm_num)\n\n/-- [PROVED session 012] √100000 ≥ 316 (316² = 99856). -/\ntheorem sqrt100000_ge : (316 : ℝ) ≤ Real.sqrt 100000 :=\n  (Real.le_sqrt (by norm_num) (by norm_num)).mpr (by norm_num)"},"extra_content":null},{"type":"tool_use","id":"Edit_165","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean","old_string":"  have hloglogT : Real.log (Real.log (Real.sqrt 100000 + 1)) ≤ (1.77 : ℝ) := by\n    have hlogTpos : (0 : ℝ) < Real.log (Real.sqrt 100000 + 1) :=\n      Real.log_pos (by linarith [Real.sqrt_nonneg 100000])\n    have h := Real.log_le_log hlogTpos (le_trans hlogT h177big)\n    rwa [Real.log_exp] at h\n  have hloglogT2 : Real.log (Real.log (Real.sqrt 100000 + 1 + 2)) ≤ (1.77 : ℝ) := by\n    have hlogT2pos : (0 : ℝ) < Real.log (Real.sqrt 100000 + 1 + 2) :=\n      Real.log_pos (by linarith [Real.sqrt_nonneg 100000])\n    have h := Real.log_le_log hlogT2pos (le_trans hlogT2 h177big)\n    rwa [Real.log_exp] at h","new_string":"  have hloglogT : Real.log (Real.log (Real.sqrt 100000 + 1)) ≤ (1.77 : ℝ) := by\n    have hlogTpos : (0 : ℝ) < Real.log (Real.sqrt 100000 + 1) :=\n      Real.log_pos (by linarith [sqrt100000_ge])\n    have h := Real.log_le_log hlogTpos (le_trans hlogT h177big)\n    rwa [Real.log_exp] at h\n  have hloglogT2 : Real.log (Real.log (Real.sqrt 100000 + 1 + 2)) ≤ (1.77 : ℝ) := by\n    have hlogT2pos : (0 : ℝ) < Real.log (Real.sqrt 100000 + 1 + 2) :=\n      Real.log_pos (by linarith [sqrt100000_ge])\n    have h := Real.log_le_log hlogT2pos (le_trans hlogT2 h177big)\n    rwa [Real.log_exp] at h"},"extra_content":null},{"type":"tool_use","id":"Edit_166","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean","old_string":"  have hlogTge1 : (1 : ℝ) ≤ Real.log (Real.sqrt 100000 + 1) := by\n    have he : Real.exp 1 ≤ Real.sqrt 100000 + 1 :=\n      le_trans Real.exp_one_lt_d9.le (by\n        have h := Real.sqrt_nonneg 100000\n        norm_num : (2.7182818286 : ℝ) ≤ 317)\n    have h := Real.log_le_log (Real.exp_pos 1) he\n    rwa [Real.log_exp] at h\n  have hloglogTnn : (0 : ℝ) ≤ Real.log (Real.log (Real.sqrt 100000 + 1)) :=\n    Real.log_nonneg hlogTge1\n  have hlogT2ge1 : (1 : ℝ) ≤ Real.log (Real.sqrt 100000 + 1 + 2) := by\n    have he : Real.exp 1 ≤ Real.sqrt 100000 + 1 + 2 :=\n      le_trans Real.exp_one_lt_d9.le (by\n        have h := Real.sqrt_nonneg 100000\n        norm_num : (2.7182818286 : ℝ) ≤ 319)\n    have h := Real.log_le_log (Real.exp_pos 1) he\n    rwa [Real.log_exp] at h","new_string":"  have hlogTge1 : (1 : ℝ) ≤ Real.log (Real.sqrt 100000 + 1) := by\n    have he : Real.exp 1 ≤ Real.sqrt 100000 + 1 :=\n      le_trans Real.exp_one_lt_d9.le (by linarith [sqrt100000_ge])\n    have h := Real.log_le_log (Real.exp_pos 1) he\n    rwa [Real.log_exp] at h\n  have hloglogTnn : (0 : ℝ) ≤ Real.log (Real.log (Real.sqrt 100000 + 1)) :=\n    Real.log_nonneg hlogTge1\n  have hlogT2ge1 : (1 : ℝ) ≤ Real.log (Real.sqrt 100000 + 1 + 2) := by\n    have he : Real.exp 1 ≤ Real.sqrt 100000 + 1 + 2 :=\n      le_trans Real.exp_one_lt_d9.le (by linarith [sqrt100000_ge])\n    have h := Real.log_le_log (Real.exp_pos 1) he\n    rwa [Real.log_exp] at h"},"extra_content":null}]