[{"type":"thinking","thinking":"Still the same issue: the goal `2 - (↑(t * (1/2))).im + s.re * (1/2) = 2 + s.re * (1/2)` — the `(↑(t * (1/2))).im` — the cast of (t·(1/2)) — its .im is NOT being evaluated to 0 by my simp-only set. The term is ((↑(t·(1/2)) : ℂ)).im — ofReal_im should give 0! My simp only set at the re bullet: [Complex.add_re, Complex.mul_re, Complex.one_re, Complex.ofReal_re, Complex.I_re, Complex.I_im, mul_zero, sub_zero, zero_mul] — ofReal_im is NOT in the re-bullet set! The ↑(t·(1/2)) is a real cast; .im of a real cast = 0 via Complex.ofReal_im. Add Complex.ofReal_im to the re bullet's set.\n\nWait — where did ↑(t·(1/2)) come from? The h1 rw changed (s/2+2).re to s.re/2+2, but the LHS (w₀ + ↑c).re evaluation produced `2 - (↑(t·(1/2))).im + s.re·(1/2)`?? Confusing — whatever: adding Complex.ofReal_im to the simp set should zero it. Let me just use plain `simp` with a full set for both bullets, then ring if needed:\n```lean\n    · rw [h1]\n      simp only [Complex.add_re, Complex.mul_re, Complex.one_re, Complex.ofReal_re,\n        Complex.I_re, Complex.I_im, Complex.ofReal_im, mul_zero, sub_zero, zero_mul,\n        mul_one, add_zero]\n      ring\n```\nHmm — wait, actually the goal's `2 - (↑(t*(1/2))).im` — where does the `2 − ...im` come from? mul_re: (a·b).re = a.re·b.re − a.im·b.im: with a = ↑(t/2), b = I: a.re·0 − a.im·1 = −(↑(t/2)).im·... the `2 − (↑(t*(1/2))).im` is weird (the 2 is from... the (2:ℂ)?). Hmm — the ext_iff goal after rw [h1]: (w₀ + ↑c).re = s.re/2 + 2 where w₀ = 1 + ↑(t/2)·I: add_re: (1 + ↑(t/2)·I).re + (↑c).re = ...: (1 + ↑(t/2)·I).re = 1 + (↑(t/2)·I).re = 1 + ((↑(t/2)).re·0 − (↑(t/2)).im·1) = 1 − (↑(t/2)).im·... — anyway the (↑(t·(1/2))).im with ofReal_im → 0 ✓. Add ofReal_im to the set. And the im bullet: goal `(w₀ + ↑c).im = t/2`: add_im: (1 + ↑(t/2)·I).im + (↑c).im = (0 + (↑(t/2)·I).im) + 0: mul_im: (↑(t/2)).re·1 + (↑(t/2)).im·0 = t/2: needs ofReal_re (for (↑(t/2)).re = t/2) and ofReal_im — add both.\n\nLet me just put both ofReal_re and ofReal_im in both bullets' sets.","signature":""},{"type":"text","text":"Adding `Complex.ofReal_im` to the simp sets."},{"type":"tool_use","id":"Edit_236","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean","old_string":"    · have h1 : (s / 2 + 2).re = s.re / 2 + 2 := by simp\n      rw [h1]\n      simp only [Complex.add_re, Complex.mul_re, Complex.one_re, Complex.ofReal_re,\n        Complex.I_re, Complex.I_im, mul_zero, sub_zero, zero_mul]\n      ring\n    · have h1 : (s / 2 + 2).im = t / 2 := by simp [him]\n      rw [h1]\n      simp only [Complex.add_im, Complex.mul_im, Complex.one_im, Complex.ofReal_im,\n        Complex.I_im, Complex.ofReal_re, mul_one, add_zero]","new_string":"    · have h1 : (s / 2 + 2).re = s.re / 2 + 2 := by simp\n      rw [h1]\n      simp only [Complex.add_re, Complex.mul_re, Complex.one_re, Complex.ofReal_re,\n        Complex.I_re, Complex.I_im, Complex.ofReal_im, mul_zero, sub_zero, zero_mul]\n      ring\n    · have h1 : (s / 2 + 2).im = t / 2 := by simp [him]\n      rw [h1]\n      simp only [Complex.add_im, Complex.mul_im, Complex.one_im, Complex.ofReal_im,\n        Complex.I_im, Complex.ofReal_re, mul_one, add_zero, mul_zero]"},"extra_content":null}]