← Session 24 — ii′-2…4: swaps, (4.10) design, contour + website roadmap fixes
thinking (1320 chars)
`Complex.norm_real (r : ℝ) : ‖(r : ℂ)‖ = ‖r‖` (real norm = |r|). Use that: `rw [h, Complex.norm_real]` then ‖x‖ on ℝ = |x| via `Real.norm_eq_abs` ✓. Let me fix all remaining items:
1. hD_rel: add `; ring`... wait — after `rw [hD_def, Complex.normSq_apply s]`, goal `s.re * s.re + s.im * s.im = s.re ^ 2 + s.im ^ 2` — rw auto-tries rfl, fails to close, leaves goal — the error shows "unsolved goals" at 242:43. Fix: change the target to mul form: `hD_rel : D = s.re * s.re + s.im * s.im` — then rw closes by rfl ✓. But nlinarith [hD_rel] uses it with sq forms elsewhere — nlinarith ring-normalizes everything, fine. But ALSO — hD_def rw direction: goal `D = s.re*s.re + s.im*s.im`; rw [hD_def] : replaces D with normSq s — wait hD_def : D = normSq s — rw [hD_def] rewrites occurrences of D (LHS of hD_def) with normSq s (RHS) ✓ goal: normSq s = s.re*s.re+s.im*s.im; rw [Complex.normSq_apply s] ✓ closes by rfl if normSq_apply's RHS is exactly `s.re * s.re + s.im * s.im` ✓.
But careful — hD_rel's statement line must match. Change statement to mul form.
Also `sq_nonneg` uses ^2 — mixing fine for nlinarith.
2. Fix sqrt-lemma leftovers: div_le_div_left → div_le_div_iff_of_pos_left, pow_le_pow_left → pow_le_pow_left₀ (two occurrences), div_eq_div_iff args (≠ 0 forms).
3. norm_ofReal → norm_real + Real.norm_eq_abs.
`Complex.norm_real` it is. Fixing all remaining items.
tool: Edit
input
{
"replace_all": false,
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean",
"old_string": " have hD_rel : D = s.re ^ 2 + s.im ^ 2 := by rw [hD_def, Complex.normSq_apply s]",
"new_string": " have hD_rel : D = s.re * s.re + s.im * s.im := by rw [hD_def, Complex.normSq_apply s]"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`Complex.norm_real (r : ℝ) : ‖(r : ℂ)‖ = ‖r‖` (real norm = |r|). Use that: `rw [h, Complex.norm_real]` then ‖x‖ on ℝ = |x| via `Real.norm_eq_abs` ✓. Let me fix all remaining items:\n\n1. hD_rel: add `; ring`... wait — after `rw [hD_def, Complex.normSq_apply s]`, goal `s.re * s.re + s.im * s.im = s.re ^ 2 + s.im ^ 2` — rw auto-tries rfl, fails to close, leaves goal — the error shows \"unsolved goals\" at 242:43. Fix: change the target to mul form: `hD_rel : D = s.re * s.re + s.im * s.im` — then rw closes by rfl ✓. But nlinarith [hD_rel] uses it with sq forms elsewhere — nlinarith ring-normalizes everything, fine. But ALSO — hD_def rw direction: goal `D = s.re*s.re + s.im*s.im`; rw [hD_def] : replaces D with normSq s — wait hD_def : D = normSq s — rw [hD_def] rewrites occurrences of D (LHS of hD_def) with normSq s (RHS) ✓ goal: normSq s = s.re*s.re+s.im*s.im; rw [Complex.normSq_apply s] ✓ closes by rfl if normSq_apply's RHS is exactly `s.re * s.re + s.im * s.im` ✓.\n\nBut careful — hD_rel's statement line must match. Change statement to mul form.\n\nAlso `sq_nonneg` uses ^2 — mixing fine for nlinarith.\n\n2. Fix sqrt-lemma leftovers: div_le_div_left → div_le_div_iff_of_pos_left, pow_le_pow_left → pow_le_pow_left₀ (two occurrences), div_eq_div_iff args (≠ 0 forms).\n\n3. norm_ofReal → norm_real + Real.norm_eq_abs.","signature":""},{"type":"text","text":"`Complex.norm_real` it is. Fixing all remaining items."},{"type":"tool_use","id":"Edit_95","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean","old_string":" have hD_rel : D = s.re ^ 2 + s.im ^ 2 := by rw [hD_def, Complex.normSq_apply s]","new_string":" have hD_rel : D = s.re * s.re + s.im * s.im := by rw [hD_def, Complex.normSq_apply s]"},"extra_content":null}]