[{"type":"thinking","thinking":"Now I can see the full block. The issues:\n- hRmin's h5 block (lines 3230-3244): the garbled h6/h7/h8 and the MulLeftStrictMono failure.\n- hre_neg (3246-3271): `linarith [h10, h14, hsign, h15]` at the end — hsign is out of scope (it's inside the by_cases in h15) — error at 3257/3269 (the h14 : (1.75)*2 ≤ s.re from hre2 — hre2 : (s/2).re < 1.75 WITHOUT rw [hre]: (s/2).re < 1.75 does NOT give s.re/2 < 1.75 unless hre is applied... in hre_neg's hsign branch: `have h14 : (1.75 : ℝ) * 2 ≤ s.re := by linarith [hre2]` — WRONG DIRECTION: hre2 : (s/2).re < 1.75 can't give a LOWER bound on s.re! That's the linarith failure at 3257. My logic: if s.re ≥ 0 then |s.re| = s.re and R−s.re < 100/R ⟹ s.re > R−100/R (large) ⟹ s.re/2 large ≥ 1.75 — contradicting hre2. The `have h14 := ... linarith [hre2]` is bogus. Rewrite per my clean design.\n- hlt (3289-3300): h13's positive branch `have h14 : (1.75)*2 ≤ s.re := by linarith [hre2]` — same bogus line + the final `linarith [h13, h14]` wrong.\n- h4 (3305-3317): the h6/h7/h8 mess — replace with the clean version.\n\nLet me rewrite lines 3220-3322 (hRmin through hdist's end) with the clean versions.","signature":""},{"type":"text","text":"Rewriting the whole block (hRmin → hdist) with the clean structure."},{"type":"tool_use","id":"Edit_186","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"      have hRmin : R - |s.re| < 100 / R := by\n        have h1 : (R - |s.re|) * (R + |s.re|) = s.im ^ 2 := by\n          have h2 : (R - |s.re|) * (R + |s.re|) = R ^ 2 - |s.re| ^ 2 := by ring\n          rw [h2, sq_abs]\n          linarith [hre2sq]\n        have h2 : R + |s.re| ≥ R := by\n          have h3 : (0 : ℝ) ≤ |s.re| := abs_nonneg _\n          linarith\n        have h3 : (R - |s.re|) * (R + |s.re|) < 100 := by linarith [h1, himsq]\n        have h4 : (0 : ℝ) < R + |s.re| := by linarith [hRpos, abs_nonneg s.re]\n        have h5 : (R - |s.re|) * (R + |s.re|) ≥ (R - |s.re|) * R :=\n          mul_le_mul_of_nonneg_left h2 (by\n            have h6 := abs_norm_sub_norm_le s (1 : ℂ)\n            rw [hs, norm_one] at h6\n            have h7 : R - 1 ≤ |R - 1| := le_abs_self _\n            have h8 : |R - ‖(1 : ℂ)‖| ≤ ‖s - (1 : ℂ)‖ := h6\n            -- R − |re s| ≥ 0 since |re s| ≤ R\n            have h9 : |s.re| ≤ R := by\n              have h10 := abs_re_le_norm s\n              rw [hs] at h10\n              exact h10\n            linarith [h9])\n        have h6 : (R - |s.re|) * R < 100 := by linarith [h3, h5]\n        exact (mul_lt_mul_right hRpos).mp (by linarith [h6] :\n          (R - |s.re|) * R < 100)\n      -- re(s/2) + R/2 ∈ [0, 50/R)\n      have hre_neg : (s / 2).re < 0 := by\n        have h9 : |s.re| ≤ R := by\n          have h10 := abs_re_le_norm s\n          rw [hs] at h10\n          exact h10\n        have h10 : |s.re| > R - 100 / R := by\n          have h11 : (0 : ℝ) ≤ |s.re| := abs_nonneg _\n          have h12 : R - |s.re| < 100 / R := hRmin\n          have h13 : |s.re| ≥ 0 := h11\n          by_cases hsign : 0 ≤ s.re\n          · rw [abs_of_nonneg hsign]\n            have h14 : (1.75 : ℝ) * 2 ≤ s.re := by linarith [hre2]\n            linarith [h14, hR]\n          · push Not at hsign\n            rw [abs_of_neg hsign]\n            linarith [h12]\n        have h14 : (100 : ℝ) / R < 1 := by\n          rw [div_lt_one hRpos]\n          linarith [hR]\n        rw [hre]\n        have h15 : s.re < 0 := by\n          by_cases hsign : 0 ≤ s.re\n          · have h16 : (1.75 : ℝ) * 2 ≤ s.re := by linarith [hre2]\n            linarith [h16, hR, h14]\n          · exact lt_of_not_ge hsign\n        linarith [h10, h14, hsign, h15]\n      -- the distance bound\n      have hdist : ∀ m : ℕ, (1 / 8 : ℝ) ≤ ‖(s / 2) + (m : ℂ)‖ := by\n        intro m\n        have h1 : ‖(s / 2) + (m : ℂ)‖ ≥ |((s / 2) + (m : ℂ)).re| := Complex.abs_re_le_norm _\n        have h2 : ((s / 2) + (m : ℂ)).re = s.re / 2 + (m : ℝ) := by\n          rw [Complex.add_re, hre, Complex.natCast_re]\n        rw [h2] at h1\n        -- (re s)/2 + R/2 ∈ [0, 50/R)\n        have hge : (0 : ℝ) ≤ s.re / 2 + R / 2 := by\n          have h9 : |s.re| ≤ R := by\n            have h10 := abs_re_le_norm s\n            rw [hs] at h10\n            exact h10\n          have h11 : -R ≤ s.re := by\n            have h12 := abs_le.mp (show |s.re| ≤ R from h9)\n            linarith [h12.1]\n          linarith [h11]\n        have hlt : s.re / 2 + R / 2 < 50 / R := by\n          have h13 : |s.re| > R - 100 / R := by\n            by_cases hsign : 0 ≤ s.re\n            · rw [abs_of_nonneg hsign]\n              have h14 : (1.75 : ℝ) * 2 ≤ s.re := by linarith [hre2]\n              linarith [h14, hR]\n            · push Not at hsign\n              rw [abs_of_neg hsign]\n              have h15 : s.re / 2 < 0 := hre_neg\n              linarith [h15, hRmin]\n          have h14 : s.re < 0 := hre_neg\n          linarith [h13, h14]\n        -- |x + m| ≥ |m − R/2| − (x + R/2) ≥ 1/4 − 50/R ≥ 1/8\n        have hgap' := hgapInt (m : ℤ)\n        have hcast : ((m : ℤ) : ℝ) = (m : ℝ) := by push_cast; ring\n        rw [hcast] at hgap'\n        have h4 : |s.re / 2 + (m : ℝ)| ≥ |(m : ℝ) - R / 2| - (s.re / 2 + R / 2) := by\n          have h5 : s.re / 2 + (m : ℝ) = ((m : ℝ) - R / 2) + (s.re / 2 + R / 2) := by ring\n          rw [h5]\n          have h6 : |(m : ℝ) - R / 2| ≤ |(m : ℝ) - R / 2 + (s.re / 2 + R / 2)|\n              + |s.re / 2 + R / 2| := by\n            have h7 := abs_add_le ((m : ℝ) - R / 2) (s.re / 2 + R / 2)\n            rw [show (m : ℝ) - R / 2 + (s.re / 2 + R / 2)\n                = (m : ℝ) - R / 2 + (s.re / 2 + R / 2) from rfl] at h7\n            have h8 : ((m : ℝ) - R / 2) = ((m : ℝ) - R / 2 + (s.re / 2 + R / 2))\n                - (s.re / 2 + R / 2) := by ring\n            rw [h8]\n            exact abs_sub_abs_le_abs_sub _ _\n          linarith [h6]\n        have h50 : (50 : ℝ) / R ≤ 1 / 8 := by\n          rw [div_le_iff₀ hRpos]\n          linarith [hR]\n        have h9 : |R / 2 - (m : ℝ)| = |(m : ℝ) - R / 2| := abs_sub_comm _ _\n        linarith [h1, h4, hgap', h9, hge, hlt, h50]","new_string":"      have hRmin : R - |s.re| < 100 / R := by\n        have h9 : |s.re| ≤ R := by\n          have h10 := abs_re_le_norm s\n          rw [hs] at h10\n          exact h10\n        have h1 : (R - |s.re|) * (R + |s.re|) = s.im ^ 2 := by\n          have h2 : (R - |s.re|) * (R + |s.re|) = R ^ 2 - |s.re| ^ 2 := by ring\n          rw [h2, sq_abs]\n          linarith [hre2sq]\n        have h3 : (R - |s.re|) * (R + |s.re|) < 100 := by linarith [h1, himsq]\n        have h5 : (R - |s.re|) * R ≤ (R - |s.re|) * (R + |s.re|) :=\n          mul_le_mul_of_nonneg_left (by linarith [abs_nonneg s.re] : R ≤ R + |s.re|)\n            (by linarith [h9] : (0 : ℝ) ≤ R - |s.re|)\n        have h6 : (R - |s.re|) * R < 100 := by linarith [h3, h5]\n        exact (div_lt_iff₀ hRpos).mpr h6\n      -- re(s/2) + R/2 ∈ [0, 50/R)\n      have hre_neg : (s / 2).re < 0 := by\n        have h14 : (100 : ℝ) / R < 1 := by\n          rw [div_lt_one hRpos]\n          linarith [hR]\n        have h15 : s.re < 0 := by\n          by_contra hcon\n          push Not at hcon\n          have h17 : |s.re| = s.re := abs_of_nonneg hcon\n          have h18 : R - s.re < 100 / R := by rw [← h17]; exact hRmin\n          rw [hre] at hre2\n          linarith [h18, hR, h14, hre2]\n        rw [hre]\n        linarith [h15]\n      -- the distance bound\n      have hdist : ∀ m : ℕ, (1 / 8 : ℝ) ≤ ‖(s / 2) + (m : ℂ)‖ := by\n        intro m\n        have h1 : ‖(s / 2) + (m : ℂ)‖ ≥ |((s / 2) + (m : ℂ)).re| := Complex.abs_re_le_norm _\n        have h2 : ((s / 2) + (m : ℂ)).re = s.re / 2 + (m : ℝ) := by\n          rw [Complex.add_re, hre, Complex.natCast_re]\n        rw [h2] at h1\n        -- (re s)/2 + R/2 ∈ [0, 50/R)\n        have hge : (0 : ℝ) ≤ s.re / 2 + R / 2 := by\n          have h9 : |s.re| ≤ R := by\n            have h10 := abs_re_le_norm s\n            rw [hs] at h10\n            exact h10\n          have h11 : -R ≤ s.re := by\n            have h12 := abs_le.mp (show |s.re| ≤ R from h9)\n            linarith [h12.1]\n          linarith [h11]\n        have hlt : s.re / 2 + R / 2 < 50 / R := by\n          have h15 : s.re < 0 := by\n            have h16 := hre_neg\n            rw [hre] at h16\n            linarith [h16]\n          have h19 : R + s.re < 100 / R := by\n            have h20 : |s.re| = -s.re := abs_of_neg h15\n            have h21 : R - (-s.re) < 100 / R := by rw [← h20]; exact hRmin\n            linarith [h21]\n          linarith [h19]\n        -- |x + m| ≥ |m − R/2| − (x + R/2) ≥ 1/4 − 50/R ≥ 1/8\n        have hgap' := hgapInt (m : ℤ)\n        have hcast : ((m : ℤ) : ℝ) = (m : ℝ) := by push_cast; ring\n        rw [hcast] at hgap'\n        have h4 : |(m : ℝ) - R / 2| - (s.re / 2 + R / 2) ≤ |s.re / 2 + (m : ℝ)| := by\n          have h5 : |(s.re / 2 + (m : ℝ)) - (s.re / 2 + R / 2)| = |(m : ℝ) - R / 2| := by\n            have h6 : (s.re / 2 + (m : ℝ)) - (s.re / 2 + R / 2) = (m : ℝ) - R / 2 := by ring\n            rw [h6]\n          have h7 := abs_sub_abs_le_abs_sub (s.re / 2 + (m : ℝ)) (s.re / 2 + R / 2)\n          rw [h5] at h7\n          have h8 : |s.re / 2 + R / 2| = s.re / 2 + R / 2 := abs_of_nonneg hge\n          linarith [h7, h8]\n        have h50 : (50 : ℝ) / R ≤ 1 / 8 := by\n          rw [div_le_iff₀ hRpos]\n          linarith [hR]\n        have h9 : |R / 2 - (m : ℝ)| = |(m : ℝ) - R / 2| := abs_sub_comm _ _\n        linarith [h1, h4, hgap', h9, hlt, h50]"},"extra_content":null}]