[{"type":"thinking","thinking":"Two linarith failures (both nlinarith spots I wrote as nlinarith? no — they're linarith calls failing on nonlinear goals):\n\n1. Line 1860 (hper's final): goal `6 * A0count * Real.log(3·2^{j+1}+5) * (1 + 3/2^j) ≤ 30 * A0count * Real.log R` — nonlinear (products of A0·log·(1+3/2^j)). linarith can't multiply hypotheses. nlinarith needed with the right product hints, or decompose: \n   LHS = 6·A0·log'·(1+3/2^j) where log' ≤ 2L (hlogle), (1+3/2^j) ≤ 5/2 (h2), all nonneg. \n   = 6·A0·log'·u with u ≤ 5/2: bound stepwise:\n   6·A0·log'·u ≤ 6·A0·(2L)·(5/2) = 30·A0·L. \n   mul_le_mul chain: (6·A0·log')·u ≤ (6·A0·2L)·(5/2): mul_le_mul (6A0log' ≤ 6A0·2L) (u ≤ 5/2) (0 ≤ u) (0 ≤ 6A0·2L):\n   - 6A0·log' ≤ 6A0·2L: mul_le_mul_of_nonneg_left hlogle (mul_nonneg (by norm_num) hA0)... 6·A0 ≥ 0: mul_nonneg (by norm_num) hA0.\n   - u = 1+3/2^j ≤ 5/2: from h2: 3/2^j ≤ 3/2 ⟹ 1+3/2^j ≤ 5/2 ✓ linarith.\n   - 0 ≤ u: 1+3/2^j ≥ 1 > 0 ✓.\n   - 0 ≤ 6A0·2L ✓.\n   Then 6·A0·2L·(5/2) = 30·A0·L ✓ by ring.\n   Rewrite with explicit mul_le_mul instead of nlinarith:\n   ```\n   have h3 : (6 : ℝ) * A0count * Real.log (3 * (2:ℝ)^(j+1) + 5) ≤ 6 * A0count * (2 * Real.log R) :=\n     mul_le_mul_of_nonneg_left hlogle (mul_nonneg (by norm_num) hA0)\n   have h4 : (1 : ℝ) ≤ 1 + 3 / (2:ℝ)^j := by linarith [div_nonneg (by norm_num : (0:ℝ) ≤ 3) h2j0.le]\n   have h5 : (1 : ℝ) + 3 / (2:ℝ)^j ≤ 5 / 2 := by linarith [h2]\n   have h6 : (6 * A0count * Real.log (3 * (2:ℝ)^(j+1) + 5)) * (1 + 3 / (2:ℝ)^j)\n       ≤ (6 * A0count * (2 * Real.log R)) * (5 / 2) :=\n     mul_le_mul h3 h5 (by linarith [h4] : (0:ℝ) ≤ 1 + 3/(2:ℝ)^j)\n       (mul_nonneg (mul_nonneg (by norm_num) hA0) (mul_nonneg (by norm_num) hlogRnn))\n   -- then (6·A0·2L)·(5/2) = 30·A0·L by ring\n   calc ... ≤ (6·A0·2L)·(5/2) := h6\n     _ = 30·A0·L := by ring\n   ```\n   The goal after rw [h1]: `6 * A0count * Real.log(...) * (1 + 3 / 2 ^ j) ≤ 30 * A0count * Real.log R` ✓ matches.\n\n2. Line 1874 (final step): `(J:ℝ)·(30·A0·L) ≤ 60·A0·L²`: J ≤ 2L, all nonneg: mul_le_mul (hJle) (le_refl (30·A0·L)) hmm: J·(30A0L) ≤ (2L)·(30A0L) = 60A0L²: mul_le_mul_of_nonneg_right hJle (0 ≤ 30A0L): (mul_nonneg (by norm_num) (mul_nonneg hA0 hlogRnn)). Then (2L)·(30A0L) = 60A0L² by ring (L² = (log R)^2: the goal has (Real.log R)^2: ring handles ^2 ✓).\n   ```\n   _ ≤ 60 * A0count * (Real.log R) ^ 2 := by\n     have h1 : (J : ℝ) * (30 * A0count * Real.log R) ≤ (2 * Real.log R) * (30 * A0count * Real.log R) :=\n       mul_le_mul_of_nonneg_right hJle (mul_nonneg (by norm_num : (0:ℝ) ≤ 30) (mul_nonneg hA0 hlogRnn))\n     have h2 : (2 * Real.log R) * (30 * A0count * Real.log R) = 60 * A0count * (Real.log R) ^ 2 := by ring\n     rw [h2] at h1\n     exact h1\n   ```\n\nLet me rewrite those two spots.","signature":""},{"type":"text","text":"Two nonlinear spots need explicit `mul_le_mul` chains instead of linarith."},{"type":"tool_use","id":"Edit_83","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"    have h2 : 3 / (2 : ℝ) ^ j ≤ 3 / 2 := by\n      apply div_le_div_of_nonneg_left (by norm_num : (0 : ℝ) ≤ 3) (by norm_num) hj2\n    have hlognn : (0 : ℝ) ≤ Real.log (3 * (2 : ℝ) ^ (j + 1) + 5) := by\n      apply Real.log_nonneg\n      have h3 : (0 : ℝ) < (2 : ℝ) ^ (j + 1) := by positivity\n      linarith [h3]\n    nlinarith [hlogle, h2, hlognn, hA0, hlogRnn]","new_string":"    have h2 : 3 / (2 : ℝ) ^ j ≤ 3 / 2 := by\n      apply div_le_div_of_nonneg_left (by norm_num : (0 : ℝ) ≤ 3) (by norm_num) hj2\n    have hlognn : (0 : ℝ) ≤ Real.log (3 * (2 : ℝ) ^ (j + 1) + 5) := by\n      apply Real.log_nonneg\n      have h3 : (0 : ℝ) < (2 : ℝ) ^ (j + 1) := by positivity\n      linarith [h3]\n    have h3 : (6 : ℝ) * A0count * Real.log (3 * (2 : ℝ) ^ (j + 1) + 5)\n        ≤ 6 * A0count * (2 * Real.log R) :=\n      mul_le_mul_of_nonneg_left hlogle (mul_nonneg (by norm_num) hA0)\n    have h4 : (0 : ℝ) ≤ 1 + 3 / (2 : ℝ) ^ j := by\n      have h5 : (0 : ℝ) ≤ 3 / (2 : ℝ) ^ j := div_nonneg (by norm_num) h2j0.le\n      linarith [h5]\n    have h5 : (1 : ℝ) + 3 / (2 : ℝ) ^ j ≤ 5 / 2 := by linarith [h2]\n    have h6 : (6 : ℝ) * A0count * Real.log (3 * (2 : ℝ) ^ (j + 1) + 5) * (1 + 3 / (2 : ℝ) ^ j)\n        ≤ (6 * A0count * (2 * Real.log R)) * (5 / 2) :=\n      mul_le_mul h3 h5 h4\n        (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 6) hA0)\n          (mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hlogRnn))\n    have h7 : (6 : ℝ) * A0count * (2 * Real.log R) * (5 / 2) = 30 * A0count * Real.log R := by\n      ring\n    rw [h7] at h6\n    exact h6"},"extra_content":null}]