[{"type":"thinking","thinking":"Found: `abs_add_le` exists in Mathlib/Algebra/Order/AbsoluteValue/Basic.lean (used at line 274 as `add_le' := abs_add_le`) — it's likely `theorem abs_add_le (a b : α) : |a + b| ≤ |a| + |b|`. Let me use `abs_add_le` instead of `abs_add`. And `div_le_iff₀` instead of div_le_iff. Both confirmed.\n\nNow let me fix:\n1. hfre with nth_rewrite 1.\n2. hA with div_le_iff₀.\n3. hB with div_le_iff₀.\n4. Final calc step with div_le_iff₀ and abs_add_le.\n\nRewrite the tail section of re_li_summand_le.","signature":""},{"type":"text","text":"`abs_add_le` and `div_le_iff₀` are the right names. Applying the final fixes."},{"type":"tool_use","id":"Edit_125","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean","old_string":"  have hfre : (1 - w ^ n).re = n * z.re / ‖z‖ ^ 2 + ((S - n) / z).re := by\n    have hd : (1 / z) * ((n : ℂ) + (S - n)) = (n : ℂ) * (1 / z) + (S - n) / z := by\n      field_simp; ring\n    rw [hf, hS, hd, Complex.add_re, hre1]\n  rw [hfre]\n  have hA : |n * z.re / ‖z‖ ^ 2| ≤ n / ‖z‖ ^ 2 := by\n    rw [abs_div, abs_mul, abs_of_nonneg hn0.le,\n      abs_of_nonneg (sq_nonneg ‖z‖)]\n    apply (div_le_div_right (by positivity)).mpr\n    apply mul_le_mul_of_nonneg_left _ hn0.le\n    exact abs_le.mpr ⟨hzre0, hzre1⟩\n  have hB : |((S - n) / z).re| ≤ n ^ 2 / ‖z‖ ^ 2 := by\n    have h1 : |((S - n) / z).re| ≤ ‖(S - n) / z‖ := Complex.abs_re_le_norm _\n    have h2 : ‖(S - n) / z‖ ≤ (n ^ 2 / ‖z‖) / ‖z‖ := by\n      rw [norm_div]\n      exact (div_le_div_right hzn).mpr hSn\n    exact h1.trans (h2.trans (by rw [div_div, ← pow_two]))\n  have hnn : (n : ℝ) ≤ n ^ 2 := by\n    have h : (1 : ℝ) ≤ n := by exact_mod_cast hn\n    nlinarith\n  calc |n * z.re / ‖z‖ ^ 2 + ((S - n) / z).re|\n      ≤ n / ‖z‖ ^ 2 + n ^ 2 / ‖z‖ ^ 2 := (abs_add _ _).trans (add_le_add hA hB)\n    _ = (n + n ^ 2) / ‖z‖ ^ 2 := by ring\n    _ ≤ 2 * n ^ 2 / ‖z‖ ^ 2 := by\n        apply (div_le_div_right (by positivity)).mpr\n        linarith","new_string":"  have hfre : (1 - w ^ n).re = n * z.re / ‖z‖ ^ 2 + ((S - n) / z).re := by\n    have hd : (1 / z) * ((n : ℂ) + (S - n)) = (n : ℂ) * (1 / z) + (S - n) / z := by\n      field_simp; ring\n    rw [hf]\n    nth_rewrite 1 [show S = (n : ℂ) + (S - n) from by ring]\n    rw [hd, Complex.add_re, hre1]\n  rw [hfre]\n  have hA : |n * z.re / ‖z‖ ^ 2| ≤ n / ‖z‖ ^ 2 := by\n    have e : |n * z.re / ‖z‖ ^ 2| = n * |z.re| / ‖z‖ ^ 2 := by\n      rw [abs_div, abs_mul, abs_of_nonneg hn0.le,\n        abs_of_nonneg (sq_nonneg ‖z‖)]\n    rw [e, div_le_iff₀ (by positivity)]\n    have e2 : (n / ‖z‖ ^ 2) * ‖z‖ ^ 2 = n := by field_simp\n    rw [e2]\n    exact mul_le_mul_of_nonneg_left (abs_le.mpr ⟨hzre0, hzre1⟩) hn0.le\n  have hB : |((S - n) / z).re| ≤ n ^ 2 / ‖z‖ ^ 2 := by\n    have h1 : |((S - n) / z).re| ≤ ‖(S - n) / z‖ := Complex.abs_re_le_norm _\n    have h2 : ‖(S - n) / z‖ ≤ (n ^ 2 / ‖z‖) / ‖z‖ := by\n      rw [norm_div, div_le_iff₀ hzn]\n      have e : ((n ^ 2 / ‖z‖) / ‖z‖) * ‖z‖ = n ^ 2 / ‖z‖ := by field_simp\n      rw [e]\n      exact hSn\n    exact h1.trans (h2.trans (by rw [div_div, ← pow_two]))\n  have hnn : (n : ℝ) ≤ n ^ 2 := by\n    have h : (1 : ℝ) ≤ n := by exact_mod_cast hn\n    nlinarith\n  calc |n * z.re / ‖z‖ ^ 2 + ((S - n) / z).re|\n      ≤ n / ‖z‖ ^ 2 + n ^ 2 / ‖z‖ ^ 2 := (abs_add_le _ _).trans (add_le_add hA hB)\n    _ = (n + n ^ 2) / ‖z‖ ^ 2 := by ring\n    _ ≤ 2 * n ^ 2 / ‖z‖ ^ 2 := by\n        rw [div_le_iff₀ (by positivity)]\n        have e : (2 * n ^ 2 / ‖z‖ ^ 2) * ‖z‖ ^ 2 = 2 * n ^ 2 := by field_simp\n        rw [e]\n        linarith"},"extra_content":null}]