[{"type":"thinking","thinking":"The route works: rw [RectangleBorder] + simp only [Set.mem_union, Complex.mem_reProdIm, Set.mem_singleton_iff] at hs + tauto, then rcases on the resulting Eq-disjunction keeps the Eqs as hypotheses.\n\nNow rewrite the hborder proof (and h1notb) in LiAssembly using this pattern. The structure per border case:\n- key : s.im = z.im ∨ s.re = z.re ∨ s.im = w.im ∨ s.re = w.re.\n- Then rcases key with the four Eq cases, each giving an Eq I can rw at the classification or use directly.\n\nRewrite the hborder proof with this pattern, plus h1notb (where s = 1: key : (1:ℂ).im = z.im ∨ (1:ℂ).re = z.re ∨ ... — each leading to a numeric contradiction). Let me rewrite those two blocks.","signature":""},{"type":"text","text":"The route works. Rewriting the border decomposition in LiAssembly with this pattern."},{"type":"tool_use","id":"Edit_281","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean","old_string":"  have h1notb : (1 : ℂ) ∉ RectangleBorder z w := by\n    intro hb\n    have h1re : (1 : ℂ).re = 1 := Complex.one_re\n    have h1im : (1 : ℂ).im = 0 := Complex.one_im\n    have h2n : (1 : ℝ) < 2 * Real.sqrt n := by linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1 : ℝ) ≤ n)]\n    rcases hb with hb | hb | hb | hb\n    · have h := Set.mem_singleton_iff.mp hb.2\n      rw [rectZ_im, Complex.one_im] at h\n      exact absurd h (ne_of_gt (by linarith [hT1] : (0 : ℝ) > -T))\n    · have h := Set.mem_singleton_iff.mp hb.1\n      rw [rectZ_re, Complex.one_re] at h\n      norm_num at h\n    · have h := Set.mem_singleton_iff.mp hb.2\n      rw [rectW_im, Complex.one_im] at h\n      exact absurd h (ne_of_gt (by linarith [hT1] : (0 : ℝ) < T))\n    · have h := Set.mem_singleton_iff.mp hb.1\n      rw [rectW_re, Complex.one_re] at h\n      exact absurd h (ne_of_gt h2n)","new_string":"  have h1notb : (1 : ℂ) ∉ RectangleBorder z w := by\n    intro hb\n    have h2n : (1 : ℝ) < 2 * Real.sqrt n := by\n      linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1 : ℝ) ≤ n)]\n    have key : (1 : ℂ).im = z.im ∨ (1 : ℂ).re = z.re ∨ (1 : ℂ).im = w.im ∨\n        (1 : ℂ).re = w.re := by\n      rw [RectangleBorder] at hb\n      simp only [Set.mem_union, Complex.mem_reProdIm, Set.mem_singleton_iff] at hb\n      tauto\n    rcases key with him | hre | him2 | hre2\n    · rw [rectZ_im, Complex.one_im] at him\n      exact absurd him (ne_of_gt (by linarith [hT1] : (0 : ℝ) > -T))\n    · rw [rectZ_re, Complex.one_re] at hre\n      norm_num at hre\n    · rw [rectW_im, Complex.one_im] at him2\n      exact absurd him2 (ne_of_gt (by linarith [hT1] : (0 : ℝ) < T))\n    · rw [rectW_re, Complex.one_re] at hre2\n      exact absurd hre2 (ne_of_gt h2n)"},"extra_content":null}]