[{"type":"thinking","thinking":"Found the current names (GroupWithZero/Basic.lean):\n- `div_le_div_iff_of_pos_right (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b` (replaces div_le_div_right)\n- `div_le_div_iff_of_pos_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b ≤ a / c ↔ c ≤ b` (replaces div_le_div_left)\n- `div_le_div_of_nonneg_left (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) : a / b ≤ a / c`\n- `pow_le_pow_left₀ [PosMulMono] [MulPosMono] (ha : 0 ≤ a) (hab : a ≤ b) (n : ℕ)?` — from the earlier grep line 470: `theorem pow_le_pow_left₀ [PosMulMono M₀] [MulPosMono M₀] (ha : 0 ≤ a) (hab : a ≤ b) (n : ℕ)? : a^n ≤ b^n` — signature probably `pow_le_pow_left₀ (ha : 0 ≤ a) (hab : a ≤ b) (n : ℕ)`.\n- `le_or_lt` — unknown; replace with `by_cases` / `lt_or_ge`: there's `lt_or_ge : a < b ∨ a ≥ b` ✓ standard. I'll use by_cases.\n\nNow fix everything:\n\n1. h1/h2 simp args: h1: `simp [Complex.add_re, Complex.one_re]`; h2: `simp [Complex.add_im, Complex.one_im]`. Wait — do those actually close the goals? The \"Try this\" suggestions said so. But hold on — the suggested h1 form leaves `(1/s).re` — how does simp get it to s.re/normSq s without div_re? There must be a default simp lemma doing it (maybe `Complex.div_re` is already @[simp]). OK trust the linter suggestions.\n\n2. hsq: restructure with set D and nlinarith/linear_combination. Also careful: after `set D`, the h1/h2/hD forms use D ✓.\n\n3. n=0 branch: simp only [kFun, pow_zero, sub_self, norm_zero] then positivity.\n\n4. `le_or_lt` → by_cases hσp : 0 ≤ s.re + 1; negative branch: not_le.mp hσp : s.re + 1 < 0.\n\n5. `pow_le_pow_left` → `pow_le_pow_left₀`.\n\n6. `div_le_div_right hD` → `(div_le_div_iff_of_pos_right hD).mpr hσ4`.\n   `div_le_div_left (by norm_num : (0:ℝ) < 4) hD (by positivity)` → `(div_le_div_iff_of_pos_left (by norm_num : (0:ℝ)<4) hD (by positivity : (0:ℝ)<(n:ℝ))).mpr hDn` — wait signature: div_le_div_iff_of_pos_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b ≤ a / c ↔ c ≤ b. I want 4/D ≤ 4/n: a=4, b=D, c=n: gives 4/D ≤ 4/n ↔ n ≤ D ✓ .mpr hDn where hDn : (n:ℝ) ≤ D ✓.\n   \n   Hmm wait — the name says \"of_pos_left\" with (ha : 0 < a) being the NUMERATOR... a/b ≤ a/c: a=4 is the shared numerator ✓.\n\n7. In kFun_shift_norm_le_exp_sqrt_half_add_one: `div_le_div_left one_pos hnorm_pos (by positivity)` → `(div_le_div_iff_of_pos_left one_pos hnorm_pos (by positivity : (0:ℝ) < 2 * Real.sqrt (n:ℝ))).mpr hnorm_s` — gives 1/‖s‖ ≤ 1/(2√n) ↔ 2√n ≤ ‖s‖ ✓.\n\nAlso `Real.sqrt_pos.mpr hnR` — Real.sqrt_pos : 0 < √x ↔ 0 < x ✓ probably fine.\n\nLet me apply all fixes.","signature":""},{"type":"text","text":"Current names found: `div_le_div_iff_of_pos_right/left`, `pow_le_pow_left₀`. Applying all fixes."},{"type":"tool_use","id":"Edit_89","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean","old_string":"theorem norm_one_add_one_div_le {s : ℂ} (hs : s ≠ 0) :\n    ‖1 + 1 / s‖ ≤ 1 + (s.re + 1) / Complex.normSq s := by\n  have hD : (0:ℝ) < Complex.normSq s := Complex.normSq_pos.mpr hs\n  have h1 : (1 + 1 / s).re = 1 + s.re / Complex.normSq s := by\n    simp [Complex.add_re, Complex.one_re, Complex.div_re, Complex.one_im]\n  have h2 : (1 + 1 / s).im = -s.im / Complex.normSq s := by\n    simp [Complex.add_im, Complex.one_im, Complex.div_im, Complex.one_re]\n  have hsq : Complex.normSq (1 + 1 / s) = 1 + (2 * s.re + 1) / Complex.normSq s := by\n    rw [Complex.normSq_apply (1 + 1 / s), h1, h2, Complex.normSq_apply s]\n    field_simp\n    ring\n  have hRHS : (0:ℝ) ≤ 1 + (s.re + 1) / Complex.normSq s := by\n    have hsum : (0:ℝ) < Complex.normSq s + (s.re + 1) := by\n      rw [Complex.normSq_apply s]\n      nlinarith [sq_nonneg (s.re + 1 / 2), sq_nonneg s.im]\n    have e : 1 + (s.re + 1) / Complex.normSq s\n        = (Complex.normSq s + (s.re + 1)) / Complex.normSq s := by\n      field_simp\n    rw [e]\n    exact div_nonneg hsum.le hD.le\n  have hcmp : ‖1 + 1 / s‖ ^ 2 ≤ (1 + (s.re + 1) / Complex.normSq s) ^ 2 := by\n    rw [← Complex.normSq_eq_norm_sq, hsq]\n    have e : (1 + (s.re + 1) / Complex.normSq s) ^ 2\n        = 1 + (2 * s.re + 1) / Complex.normSq s\n          + (1 / Complex.normSq s + ((s.re + 1) / Complex.normSq s) ^ 2) := by\n      field_simp\n      ring\n    rw [e]\n    have hpos : (0:ℝ) ≤ 1 / Complex.normSq s + ((s.re + 1) / Complex.normSq s) ^ 2 := by\n      positivity\n    linarith\n  exact (sq_le_sq₀ (norm_nonneg _) hRHS).mp hcmp","new_string":"theorem norm_one_add_one_div_le {s : ℂ} (hs : s ≠ 0) :\n    ‖1 + 1 / s‖ ≤ 1 + (s.re + 1) / Complex.normSq s := by\n  set D := Complex.normSq s with hD_def\n  have hD : (0:ℝ) < D := Complex.normSq_pos.mpr hs\n  have hD_rel : D = s.re ^ 2 + s.im ^ 2 := by rw [hD_def, Complex.normSq_apply s]\n  have h1 : (1 + 1 / s).re = 1 + s.re / D := by\n    simp [Complex.add_re, Complex.one_re, hD_def]\n  have h2 : (1 + 1 / s).im = -s.im / D := by\n    simp [Complex.add_im, Complex.one_im, hD_def]\n  have hsq : Complex.normSq (1 + 1 / s) = 1 + (2 * s.re + 1) / D := by\n    rw [Complex.normSq_apply (1 + 1 / s), h1, h2]\n    field_simp [hD.ne']\n    nlinarith [hD_rel]\n  have hRHS : (0:ℝ) ≤ 1 + (s.re + 1) / D := by\n    have hsum : (0:ℝ) < D + (s.re + 1) := by\n      nlinarith [sq_nonneg (s.re + 1 / 2), sq_nonneg s.im, hD_rel]\n    have e : 1 + (s.re + 1) / D = (D + (s.re + 1)) / D := by field_simp [hD.ne']\n    rw [e]\n    exact div_nonneg hsum.le hD.le\n  have hcmp : ‖1 + 1 / s‖ ^ 2 ≤ (1 + (s.re + 1) / D) ^ 2 := by\n    rw [← Complex.normSq_eq_norm_sq, hsq]\n    have e : (1 + (s.re + 1) / D) ^ 2\n        = 1 + (2 * s.re + 1) / D + (1 / D + ((s.re + 1) / D) ^ 2) := by\n      field_simp [hD.ne']\n      ring\n    rw [e]\n    have hpos : (0:ℝ) ≤ 1 / D + ((s.re + 1) / D) ^ 2 := by positivity\n    linarith\n  exact (sq_le_sq₀ (norm_nonneg _) hRHS).mp hcmp"},"extra_content":null}]