[{"type":"thinking","thinking":"1244:55 is `by norm_num` proving `(3.9636:ℝ) ≤ 0.8862943606 * 4.4721` — check: 0.8862943606*4.4721 = ? 0.8862943606*4 = 3.5451774424; 0.8862943606*0.4721 = 0.418419...: 0.8862943606*0.4 = 0.3545177; *0.0721 = 0.063902; total 0.418420; sum = 3.963597. So the product = 3.963597 < 3.9636!! ✗ norm_num fails (unsolved goals ⊢ False). My constant is wrong: 3.9636 > 3.963597. Use 3.9635: 3.9635 ≤ 3.963597 ✓. Then downstream: exp(3.9635) ≥ Σ₇(3.9635) — recompute the sum: x=3.9635: x² = 15.7093/2 = 7.8547; x³ = 62.256/6 = 10.376; x⁴ = 246.70/24 = 10.279; x⁵ = 977.75/120 = 8.148; x⁶ = 3875.3/720 = 5.382. Sum: 1+3.9635+7.8547+10.376+10.279+8.148+5.382 = 47.003. ≥ 47.0. So use 47.0 as the exp lower bound: 1/47 ≤ 0.0212766 — then hexp ≤ 0.0212766? Then u₂ ≤ 9.9811*0.0212766 = 0.21238 ≤ 0.2124 ✓ still fine!\n\nSo: ht : (3.9635:ℝ) ≤ a*√20 (0.8862943606*4.4721 = 3.963597 ≥ 3.9635 ✓ norm_num). hsum : (47.0:ℝ) ≤ exp(a√20) (Σ₇(3.9635) ≥ 47.0: norm_num evaluates the sum to 47.003 ≥ 47.0 ✓). Then 1/47 ≤ 0.0212766: check 0.0212766*47 = 1.0000002 ≥ 1 ✓ (nice). hexp ≤ 0.0212766. hu2val: 9.9811*0.0212766 = 0.212379 ≤ 0.2124 ✓ norm_num.\n\nAdjust: ht constant 3.9635; h1 := sum_le_exp_of_nonneg (x := 3.9635); h3 : 47.0 ≤ Σ; hsum : 47.0 ≤ exp; calc: 1/47; div_le_iff₀ (0 < 47); target 0.0212766; final mul norm_num: 9.9811*0.0212766 ≤ 0.2124.\n\nTimeouts: 1264 (nlinarith [hb] Simplex) — hpoly's nlinarith [hb] — the constants: 1.92182*4.4722 = 8.5947805; +hb's 1.3862943616: need ≤ 9.9811: 8.5947805+1.3862943616 = 9.9810749 ≤ 9.9811 ✓. The Simplex timeout might be cascade from earlier failure OR nlinarith with the big context + hb's product... linarith should do it directly (no products needed — h1 has the product as a hypothesis with folded constant). Try linarith instead of nlinarith [hb]: h1 : b²√20 ≤ 8.5947805-ish; hb : b ≤ 1.3862943616; goal b²√20 + b ≤ 9.9811: linarith ✓ linear! Change to `linarith`.\n\n1130:0 whnf timeout — at the theorem header — cascades likely; rebuild after fixes.","signature":""},{"type":"text","text":"Constant too tight again: `0.8862943606·4.4721 = 3.963597 < 3.9636`. Loosening the chain."},{"type":"tool_use","id":"Edit_239","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"      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,\n              (3.9636 : ℝ) ^ i / (Nat.factorial i : ℝ) := by\n            norm_num [Finset.sum_range_succ]\n          linarith\n        rw [show -a * Real.sqrt 20 = -(a * Real.sqrt 20) by ring, Real.exp_neg]\n        calc (Real.exp (a * Real.sqrt 20))⁻¹ ≤ 1 / 47.01 := by\n              rw [← one_div]\n              exact 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 (Real.sqrt_nonneg 20) (by norm_num)\n        nlinarith [hb]","new_string":"      have hexp : Real.exp (-a * Real.sqrt 20) ≤ (0.0212766 : ℝ) := by\n        have ht : (3.9635 : ℝ) ≤ a * Real.sqrt 20 := by\n          calc (3.9635 : 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 : (47.0 : ℝ) ≤ Real.exp (a * Real.sqrt 20) := by\n          have h1 := Real.sum_le_exp_of_nonneg (x := (3.9635 : ℝ)) (by norm_num) 7\n          have h2 : Real.exp (3.9635 : ℝ) ≤ Real.exp (a * Real.sqrt 20) :=\n            Real.exp_le_exp.mpr ht\n          have h3 : (47.0 : ℝ) ≤ ∑ i ∈ Finset.range 7,\n              (3.9635 : ℝ) ^ i / (Nat.factorial i : ℝ) := by\n            norm_num [Finset.sum_range_succ]\n          linarith\n        rw [show -a * Real.sqrt 20 = -(a * Real.sqrt 20) by ring, Real.exp_neg]\n        calc (Real.exp (a * Real.sqrt 20))⁻¹ ≤ 1 / 47 := by\n              rw [← one_div]\n              exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 47) hsum\n          _ ≤ 0.0212766 := by\n              rw [div_le_iff₀ (by norm_num : (0 : ℝ) < 47)]; 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 (Real.sqrt_nonneg 20) (by norm_num)\n        linarith"},"extra_content":null}]