[{"type":"thinking","thinking":"The file is contaminated from line 190 to the end (371 lines) with my draft. The `end NguyenRH1`/`end` markers are inside the junk or missing. Cleanest: truncate the file to line 189 (keeping through line 188 + blank 189), then append the correct §2 + the end markers via a fresh careful Write... The Edit tool needs exact old_string matching; the junk is huge. Simplest: use Bash to truncate the file to line 188, then append the §2 content via Write... but Write overwrites the whole file. Better: truncate via bash `head -188 > tmp && mv`, then use Edit to append? Edit can't append without matching... I'll truncate with head and then use Write to rewrite... no — Write replaces the whole file; I'd need to re-paste §1. Simplest reliable: truncate with head, then append §2 via a heredoc `cat >>`. But the instructions say prefer dedicated tools over bash heredocs for file creation... The Edit tool appends by matching the last line: old_string = the last good lines, new_string = same + the §2 content. That works: match on lines 186–188 (the last three lines, unique):\n```\n    rw [Rat.cast_inv, Rat.cast_natCast, one_div]\n  rw [h1]\n  exact harmonic_le_one_add_log N\n```\nand replace with itself + the new content. Let me do that — truncate first via head (to remove the junk), then Edit to append §2 + §3 + §4 + §5 + end markers.\n\nWait — the junk section might contain the `end NguyenRH1 / end` markers. Truncating to line 188 removes them; I'll re-add via the Edit append. Plan:\n1. head -188 > tmp; mv (bash).\n2. Edit: append §2 (norm_digamma_le_of_two_le_re complete) + §3 + §4 + §5 + ends.\n\nGiven the append will be LARGE, let me split into two Edits: §2 first, compile; then §3+§4+§5.\n\nActually — to minimize risk of re-pasting analysis text, let me write §2's Lean code directly as the final form, being careful.\n\n§2 final theorem structure (complete):\n```\n/-- **The right-half-plane digamma bound (B)**: ... -/\ntheorem norm_digamma_le_of_two_le_re {w : ℂ} (hwre : 2 ≤ w.re) (hwim : |w.im| ≤ 5) :\n    ‖Complex.digamma w‖ ≤ 20 * Real.log (2 + ‖w‖) := by\n  have hw_norm_ge : 2 ≤ ‖w‖ := by\n    have h1 := Complex.abs_re_le_norm w\n    rwa [abs_of_nonneg (by linarith [hwre])] at h1\n  by_cases hwint : ∃ m : ℤ, (m : ℂ) = w\n  · -- positive-integer case\n    obtain ⟨m, hm⟩ := hwint\n    have hmre : w.re = (m : ℝ) := by rw [← hm]; simp\n    have hm2 : 2 ≤ m := by\n      by_contra h\n      push Not at h\n      have h1 : (m : ℝ) ≤ 1 := by exact_mod_cast (show m ≤ 1 by omega)\n      linarith [hwre, hmre, h1]\n    set n := m.toNat with hn_def\n    have hn2 : 2 ≤ n := by\n      have h1 : (2 : ℤ).toNat ≤ m.toNat := Int.toNat_mono hm2\n      have h2 : (2 : ℤ).toNat = 2 := by decide\n      rwa [h2] at h1  -- h1 : 2 ≤ m.toNat; goal 2 ≤ n with n := m.toNat: rwa [hn_def]: hmm set replaced: goal 2 ≤ n: h1 : 2 ≤ m.toNat: rw [hn_def] at h1? set introduces hn_def : n = m.toNat: exact h1 after rw [hn_def]:\n      ...\n```\ncareful with set: after `set n := m.toNat with hn_def`, n is a local def; hn2 : 2 ≤ n: `rw [hn_def]` changes goal to 2 ≤ m.toNat... hn_def : n = m.toNat: rw [hn_def] rewrites n → m.toNat in goal ✓ then exact the h1. OK:\n```\n    have hn2 : 2 ≤ n := by\n      rw [hn_def]\n      have h1 : (2 : ℤ).toNat ≤ m.toNat := Int.toNat_mono hm2\n      have h2 : (2 : ℤ).toNat = 2 := by decide\n      rwa [h2] at h1\n    have hwn : w = (n : ℂ) := by\n      rw [← hm, hn_def]\n      exact_mod_cast Int.toNat_of_nonneg (by omega : (0 : ℤ) ≤ m)\n    rw [hwn, digamma_natCast_eq n (by omega)]\n    -- norm bound\n    have hγ0 : (0 : ℝ) ≤ Real.eulerMascheroniConstant := Real.one_half_lt_eulerMascheroniConstant.le.trans (by norm_num)\n    have hγ1 : Real.eulerMascheroniConstant ≤ 1 := Real.eulerMascheroniConstant_lt_two_thirds.le.trans (by norm_num)\n    have hH0 : (0 : ℝ) ≤ ((harmonic (n - 1) : ℚ) : ℝ) := by positivity\n    have hHbound : ((harmonic (n - 1) : ℚ) : ℝ) ≤ 1 + Real.log (n - 1) := harmonic_le_one_add_log (n - 1)\n    -- the cast form: the goal's norm: ‖-(γ:ℂ) + (↑(harmonic (n−1)):ℂ)‖: rewrite as the real abs:\n    have hnrm : ‖-(Real.eulerMascheroniConstant : ℂ) + ((harmonic (n - 1) : ℚ) : ℂ)‖\n        = |(-Real.eulerMascheroniConstant + (harmonic (n - 1) : ℚ) : ℝ)| := by\n      rw [← Complex.ofReal_neg, ← Complex.ofReal_ratCast?...]  -- the harmonic cast: ((harmonic (n−1) : ℚ) : ℂ): Rat.cast to ℂ: Complex.ofReal_ratCast? hmm — (↑q : ℂ) with q : ℚ: Complex.ofReal_ratCast : ((q : ℝ):ℂ)?? hmm the cast ℚ→ℂ directly: Rat.cast to ℂ: is ((q:ℂ)) = (((q:ℝ)):ℂ)? `Complex.ofReal_ratCast (q : ℚ) : ((q : ℝ) : ℂ) = (q : ℂ)`? hmm — there IS `Complex.ofReal_ratCast`? not sure. Alternative: push_cast handles ℚ→ℝ→ℂ chains? The mixed casts: ((harmonic (n−1) : ℚ) : ℂ) — direct ℚ→ℂ cast (Rat.cast to ℂ). And I want to relate to ((harmonic (n−1) : ℚ) : ℝ) — ℚ→ℝ. The bridge: ((q:ℂ)) = (((q:ℝ)):ℂ): `Complex.ofReal_ratCast`? Let me check: there's `Complex.ofReal_ratCast : ((q : ℝ) : ℂ) = ↑q`? hmm actually `Complex.ofReal_ratCast (q : ℚ) : (↑(↑q : ℝ)) = ↑q`? I'll check via grep. If it exists: norm: ‖(↑q : ℂ)‖ = |(↑q : ℝ)|: via ofReal_ratCast backwards + Complex.norm_real + Real.norm_eq_abs: ‖(↑q:ℂ)‖ = ‖((q:ℝ):ℂ)‖ [ofReal_ratCast] = ‖(q:ℝ)‖ [norm_real] = |q| [Real.norm_eq_abs].\n```\nThis ℚ→ℂ vs ℚ→ℝ→ℂ cast bridge needs checking. Let me grep for ofReal_ratCast.","signature":""},{"type":"text","text":"The file is contaminated from line 190 on. Truncating to the good part and checking a cast-bridge name before rewriting."},{"type":"tool_use","id":"Bash_266","name":"Bash","input":{"command":"head -188 /d/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiDigammaLeft.lean > /tmp/LiDigammaLeft_trunc.lean && mv /tmp/LiDigammaLeft_trunc.lean /d/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiDigammaLeft.lean && tail -3 /d/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiDigammaLeft.lean","description":"Truncate LiDigammaLeft.lean to the good prefix"},"extra_content":null}]