[{"type":"thinking","thinking":"I left sorries and a placeholder `(div_le_div_iff_of_pos_left??).mpr ??` — I was mid-thought on the exp bound. Let me finish hu2val and the rest cleanly. The exp bound: exp(−a√20) ≤ 0.0212721 ⟺ (exp(a√20))⁻¹ ≤ 0.0212721 ⟺ 1/exp(a√20) ≤ 0.0212721. From hsum : 47.01 ≤ exp(a√20): 1/exp(a√20) ≤ 1/47.01 (one_div_le_one_div_of_le (by norm_num : 0 < 47.01) hsum). And 1/47.01 ≤ 0.0212721: rw [div_le_iff₀ (by norm_num)]; norm_num (1 ≤ 0.0212721*47.01 = 1.00000... ✓).\n\nLet me rewrite the tail of hu2fac (from `have hexp` onward) plus hu2val's completion, then u3, final assembly. Rewrite the broken part: replace from `have hexp` to the end with correct code.","signature":""},{"type":"text","text":"I left placeholders mid-proof. Completing the numeric tail properly."},{"type":"tool_use","id":"Edit_184","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"    have hu2le : u2 s ≤ u2 (Real.sqrt 20) :=\n      hu2anti (Set.mem_Ici.mpr hsqrthi.le) (Set.mem_Ici.mpr hs)\n    have hu2val : u2 (Real.sqrt 20) ≤ (0.2124 : ℝ) := by\n      have hexp : Real.exp (-a * Real.sqrt 20) ≤ (0.0212721 : ℝ) := by\n        have ht : (3.9636 : ℝ) ≤ a * Real.sqrt 20 := by\n          calc (3.9636 : ℝ) ≤ 0.8862943606 * 4.4721 := by norm_num\n            _ ≤ a * Real.sqrt 20 :=\n              mul_le_mul ha hsqrtlo (by norm_num) hapos.le\n        have hsum : (47.01 : ℝ) ≤ Real.exp (a * Real.sqrt 20) := by\n          have h1 := Real.sum_le_exp_of_nonneg (x := (3.9636 : ℝ)) (by norm_num) 7\n          have h2 : Real.exp (3.9636 : ℝ) ≤ Real.exp (a * Real.sqrt 20) :=\n            Real.exp_le_exp.mpr ht\n          have h3 : (47.01 : ℝ) ≤ ∑ i ∈ Finset.range 7, (3.9636 : ℝ) ^ i / i ! := by\n            norm_num [Finset.sum_range_succ]\n          linarith\n        rw [Real.exp_neg, div_le_iff₀ (by norm_num : (0 : ℝ) < 47.01)] at *\n        rw [Real.exp_neg]\n        exact (div_le_div_iff_of_pos_left??).mpr ??\n      sorry\n    sorry\n  sorry","new_string":"    have hu2le : u2 s ≤ u2 (Real.sqrt 20) :=\n      hu2anti (Set.mem_Ici.mpr hsqrthi.le) (Set.mem_Ici.mpr hs)\n    have hu2val : u2 (Real.sqrt 20) ≤ (0.2124 : ℝ) := by\n      have hexp : Real.exp (-a * Real.sqrt 20) ≤ (0.0212721 : ℝ) := by\n        have ht : (3.9636 : ℝ) ≤ a * Real.sqrt 20 := by\n          calc (3.9636 : ℝ) ≤ 0.8862943606 * 4.4721 := by norm_num\n            _ ≤ a * Real.sqrt 20 :=\n              mul_le_mul ha hsqrtlo (by norm_num) hapos.le\n        have hsum : (47.01 : ℝ) ≤ Real.exp (a * Real.sqrt 20) := by\n          have h1 := Real.sum_le_exp_of_nonneg (x := (3.9636 : ℝ)) (by norm_num) 7\n          have h2 : Real.exp (3.9636 : ℝ) ≤ Real.exp (a * Real.sqrt 20) :=\n            Real.exp_le_exp.mpr ht\n          have h3 : (47.01 : ℝ) ≤ ∑ i ∈ Finset.range 7, (3.9636 : ℝ) ^ i / i ! := by\n            norm_num [Finset.sum_range_succ]\n          linarith\n        rw [Real.exp_neg]\n        calc (Real.exp (a * Real.sqrt 20))⁻¹ ≤ 1 / 47.01 :=\n              one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 47.01) hsum\n          _ ≤ 0.0212721 := by\n              rw [div_le_iff₀ (by norm_num : (0 : ℝ) < 47.01)]; norm_num\n      have hpoly : b ^ 2 * Real.sqrt 20 + b ≤ (9.9811 : ℝ) := by\n        have h1 : b ^ 2 * Real.sqrt 20 ≤ 1.92182 * 4.4722 :=\n          mul_le_mul hb2up hsqrthi (by norm_num) (sq_nonneg b)\n        nlinarith [hb]\n      calc u2 (Real.sqrt 20) = (b ^ 2 * Real.sqrt 20 + b) * Real.exp (-a * Real.sqrt 20) := rfl\n        _ ≤ 9.9811 * 0.0212721 :=\n          mul_le_mul hpoly hexp (Real.exp_pos _).le (by norm_num)\n        _ ≤ 0.2124 := by norm_num\n    calc (b ^ 2 * s + b) * eA = u2 s := rfl\n      _ ≤ u2 (Real.sqrt 20) := hu2le\n      _ ≤ 0.2124 := hu2val\n  have hu3 : 1 + Real.exp (-s / 2) ≤ (1.1099 : ℝ) := by\n    have hexp : Real.exp (-s / 2) ≤ (0.10982 : ℝ) := by\n      have ht : (2.23605 : ℝ) ≤ s / 2 := by linarith\n      have hsum : (9.1064 : ℝ) ≤ Real.exp (s / 2) := by\n        have h1 := Real.sum_le_exp_of_nonneg (x := (2.23605 : ℝ)) (by norm_num) 6\n        have h2 : Real.exp (2.23605 : ℝ) ≤ Real.exp (s / 2) := Real.exp_le_exp.mpr ht\n        have h3 : (9.1064 : ℝ) ≤ ∑ i ∈ Finset.range 6, (2.23605 : ℝ) ^ i / i ! := by\n          norm_num [Finset.sum_range_succ]\n        linarith\n      rw [Real.exp_neg]\n      calc (Real.exp (s / 2))⁻¹ ≤ 1 / 9.1064 :=\n            one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 9.1064) hsum\n        _ ≤ 0.10982 := by\n            rw [div_le_iff₀ (by norm_num : (0 : ℝ) < 9.1064)]; norm_num\n    linarith\n  -- final assembly\n  have hpos1 : (0 : ℝ) ≤ (b ^ 2 * s + b) * eA :=\n    mul_nonneg (by nlinarith [hbpos, hs0, sq_nonneg b]) heA.le\n  have hpos2 : (0 : ℝ) ≤ 1 + Real.exp (-s / 2) := by positivity\n  have hpos3 : (0 : ℝ) ≤ (s + 1) / s := by positivity\n  have hchain : 4.56 * ((s + 1) / s) * ((b ^ 2 * s + b) * eA) * (1 + Real.exp (-s / 2)) ≤\n      4.56 * 1.2237 * 0.2124 * 1.1099 := by\n    have h1 : ((s + 1) / s) * ((b ^ 2 * s + b) * eA) ≤ 1.2237 * 0.2124 :=\n      mul_le_mul hu1 hu2fac hpos1 (by norm_num)\n    have h2 : ((s + 1) / s) * ((b ^ 2 * s + b) * eA) * (1 + Real.exp (-s / 2)) ≤\n        1.2237 * 0.2124 * 1.1099 :=\n      mul_le_mul h1 hu3 hpos2 (by positivity)\n    have h3 : (0 : ℝ) ≤ 4.56 := by norm_num\n    exact mul_le_mul_of_nonneg_left h2 h3\n  calc 4.56 * (2 * (-a * eA - b * eB) + (s + 1) * (a ^ 2 * eA + b ^ 2 * eB)) -\n          (4.56 * ((eA + eB) + (s + 1) * (-a * eA - b * eB))) / s\n      ≤ 4.56 * (s + 1) * (b ^ 2 + b / s) * (eA + eB) := le_trans hstep1 hstep2\n    _ = 4.56 * ((s + 1) / s) * ((b ^ 2 * s + b) * eA) * (1 + Real.exp (-s / 2)) := hstep3\n    _ ≤ 4.56 * 1.2237 * 0.2124 * 1.1099 := hchain\n    _ < 2 := by norm_num"},"extra_content":null}]