[{"type":"thinking","thinking":"The heredoc broke — the Python source contains `?` or quote issues? Actually the error is bash-level: \"unexpected EOF while looking for matching `''\" — the heredoc content has a `'` in a string that terminated something? I used `<<'EOF'` (quoted heredoc — no shell expansion, so `'` inside is fine). Hmm, but the error says unexpected EOF looking for matching quote — the issue: my python code contains the line `show farW (k + 4 + 1) = farW (k + 5)` — no quotes issue... Actually — the heredoc is fed to `python -` via stdin: `python - <<'EOF'`. The content includes `<<'EOF'`... no. Let me look: the error is at \"line 175\" of the bash command — the heredoc itself must have been terminated early by a line containing exactly EOF? No line is exactly EOF... wait, my content has `EOF` nowhere. Hmm — maybe the issue: the previous command used `python - <<'EOF'` successfully twice. This time the content includes `«show farW (k + 4 + 1) = farW (k + 5)»` — fine. Let me check for unbalanced parens in the bash sense... heredoc shouldn't care. \n\nOH WAIT — I see: the command was `python - <<'EOF'` and inside there's a line `print(\"done\")` then `EOF` then the `export...` — but the error says line 175 unexpected EOF while looking for matching `'`. Maybe a `'` character in the Lean code strings: `⊢`? No quotes... Let me search my heredoc for single quotes: there's `'done'`... I used double quotes for print. Hmm — there are `'` in the Lean identifiers like `h2'`, `h3'`, `h4'`!! Those are inside the quoted heredoc — should be fine since `<<'EOF'` quotes everything until EOF...\n\nUnless the heredoc delimiter got messed up by the content containing a line starting with `EOF`? No.\n\nActually — maybe the issue is the Bash tool wrapping: the command had embedded newlines and the tool does its own quoting. The previous two python heredocs worked fine though. The difference: this one is much bigger (175+ lines). Perhaps a truncation issue. Let me just write the Python to a temp file with the Write tool and run it.","signature":""},{"type":"text","text":"The heredoc broke at scale. Let me write the patch script to a file instead."},{"type":"tool_use","id":"Write_77","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\patch_s10.py","content":"p = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\n# farW_le_inv_mul_sub: restrict to 2 <= k (the k=1 case is false), fix abs_of_nonneg\nold1 = \"\"\"/-- farW k ≤ 1/(k(k−1)) for k ≥ 1 (since √(1+k²) ≥ k−1). -/\nlemma farW_le_inv_mul_sub {k : ℕ} (hk : 1 ≤ k) :\n    farW k ≤ 1 / ((k : ℝ) * ((k : ℝ) - 1)) := by\n  have hk0 : (0 : ℝ) < (k : ℝ) := by exact_mod_cast hk\n  rw [farW]\n  have h1 : (k : ℝ) * ((k : ℝ) - 1) ≤ (k : ℝ) * Real.sqrt (1 + (k : ℝ) ^ 2) := by\n    apply mul_le_mul_of_nonneg_left _ hk0.le\n    by_cases hk1 : (1 : ℝ) ≤ (k : ℝ)\n    · calc (k : ℝ) - 1 ≤ (k : ℝ) := by linarith\n        _ = Real.sqrt ((k : ℝ) ^ 2) := by\n            rw [Real.sqrt_sq_eq_abs, abs_of_nonneg hk1]\n        _ ≤ Real.sqrt (1 + (k : ℝ) ^ 2) :=\n            Real.sqrt_le_sqrt (by linarith [sq_nonneg (k : ℝ)])\n    · have h2 : (k : ℝ) - 1 ≤ 0 := by linarith\n      exact h2.trans (Real.sqrt_nonneg _)\n  rcases eq_or_ne k 1 with rfl | hk1\n  · norm_num [farW]\n  · have hkk : (1 : ℝ) < (k : ℝ) := by\n      have h1' : (1 : ℕ) < k := lt_of_le_of_ne hk (Ne.symm hk1)\n      exact_mod_cast h1'\n    have h2 : (0 : ℝ) < (k : ℝ) * ((k : ℝ) - 1) := by positivity\n    exact one_div_le_one_div_of_le h2 h1\"\"\"\nnew1 = \"\"\"/-- farW k ≤ 1/(k(k−1)) for k ≥ 2 (since √(1+k²) ≥ k−1). -/\nlemma farW_le_inv_mul_sub {k : ℕ} (hk : 2 ≤ k) :\n    farW k ≤ 1 / ((k : ℝ) * ((k : ℝ) - 1)) := by\n  have hk0 : (0 : ℝ) < (k : ℝ) := by exact_mod_cast (by omega : 1 ≤ k)\n  have hk1 : (1 : ℝ) ≤ (k : ℝ) := by exact_mod_cast (by omega : 1 ≤ k)\n  rw [farW]\n  have h1 : (k : ℝ) * ((k : ℝ) - 1) ≤ (k : ℝ) * Real.sqrt (1 + (k : ℝ) ^ 2) := by\n    apply mul_le_mul_of_nonneg_left _ hk0.le\n    calc (k : ℝ) - 1 ≤ (k : ℝ) := by linarith\n      _ = Real.sqrt ((k : ℝ) ^ 2) := by\n          rw [Real.sqrt_sq_eq_abs, abs_of_nonneg hk0.le]\n      _ ≤ Real.sqrt (1 + (k : ℝ) ^ 2) :=\n          Real.sqrt_le_sqrt (by linarith [sq_nonneg (k : ℝ)])\n  have hkk : (1 : ℝ) < (k : ℝ) := by exact_mod_cast (by omega : 1 < k)\n  have h2 : (0 : ℝ) < (k : ℝ) * ((k : ℝ) - 1) := by positivity\n  exact one_div_le_one_div_of_le h2 h1\"\"\"\nassert old1 in src, \"old1 missing\"\nsrc = src.replace(old1, new1)\n\n# telescope: restructure with a standalone HasSum lemma\nold2 = \"\"\"/-- The telescoping tail: ∑' k, (1/(k+4) − 1/(k+5)) = 1/4. -/\nlemma tsum_one_div_four_sub_telescope :\n    (∑' k : ℕ, (1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5))) = 1 / 4 := by\n  have ha : ∀ k : ℕ, 1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5)\n      = 1 / ((k : ℝ) + 4) / ((k : ℝ) + 5) := by\n    intro k\n    have h1 : ((k : ℝ) + 4) ≠ 0 := by positivity\n    have h2 : ((k : ℝ) + 5) ≠ 0 := by positivity\n    field_simp\n    ring\n  have hsum : ∀ N : ℕ, ∑ k ∈ Finset.range N, (1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5))\n      = 1 / 4 - 1 / ((N : ℝ) + 4) := by\n    intro N\n    have h := Finset.sum_range_sub' (fun k : ℕ => 1 / ((k : ℝ) + 4)) N\n    rw [h]\n    simp\n  have hlim : Filter.Tendsto (fun N : ℕ => 1 / ((N : ℝ) + 4)) Filter.atTop (𝓝 0) := by\n    have h1 : Filter.Tendsto (fun N : ℕ => (N : ℝ) + 4) Filter.atTop Filter.atTop :=\n      Filter.Tendsto.atTop_add_const_right _ _ tendsto_natCast_atTop_atTop\n    have h2 := tendsto_inv_atTop_zero.comp h1\n    simpa [one_div] using h2\n  have hhas : HasSum (fun k : ℕ => 1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5)) (1 / 4) := by\n    rw [HasSum]\n    rw [show (fun N => ∑ k ∈ Finset.range N, (1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5)))\n        = fun N => 1 / 4 - 1 / ((N : ℝ) + 4) from funext hsum]\n    have h3 : (1 : ℝ) / 4 = 1 / 4 - 0 := by ring\n    rw [h3]\n    exact Filter.Tendsto.sub tendsto_const_nhds hlim\n  exact hhas.tsum_eq\"\"\"\nnew2 = \"\"\"/-- The telescoping tail HasSum: (1/(k+4) − 1/(k+5)) has sum 1/4. -/\nlemma hasSum_one_div_four_sub_telescope :\n    HasSum (fun k : ℕ => 1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5)) (1 / 4) := by\n  have hsum : ∀ N : ℕ, ∑ k ∈ Finset.range N, (1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5))\n      = 1 / 4 - 1 / ((N : ℝ) + 4) := by\n    intro N\n    have h := Finset.sum_range_sub' (fun k : ℕ => 1 / ((k : ℝ) + 4)) N\n    have h2 : ∀ i : ℕ, 1 / ((i : ℝ) + 4) - 1 / (((i + 1 : ℕ) : ℝ) + 4)\n        = 1 / ((i : ℝ) + 4) - 1 / ((i : ℝ) + 5) := by\n      intro i\n      push_cast\n      ring_nf\n    rw [Finset.sum_congr rfl h2] at h\n    have h3 : 1 / ((0 : ℝ) + 4) = 1 / 4 := by norm_num\n    rw [h3] at h\n    exact h\n  have hlim : Filter.Tendsto (fun N : ℕ => 1 / ((N : ℝ) + 4)) Filter.atTop (𝓝 0) := by\n    have h1 : Filter.Tendsto (fun N : ℕ => ((N + 4 : ℕ) : ℝ)) Filter.atTop Filter.atTop :=\n      tendsto_natCast_atTop_atTop.comp (tendsto_add_atTop_nat 4)\n    have h2 := tendsto_inv_atTop_zero.comp h1\n    have h3 : (fun N : ℕ => (((N + 4 : ℕ) : ℝ))⁻¹) = (fun N : ℕ => (1 / ((N : ℝ) + 4))) := by\n      funext N\n      push_cast\n      ring_nf\n    rw [← h3]\n    simpa [Function.comp_def] using h2\n  rw [HasSum]\n  rw [show (fun N => ∑ k ∈ Finset.range N, (1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5)))\n      = fun N : ℕ => 1 / 4 - 1 / ((N : ℝ) + 4) from funext hsum]\n  have h4 : (1 : ℝ) / 4 = 1 / 4 - 0 := by ring\n  rw [h4]\n  exact Filter.Tendsto.sub tendsto_const_nhds hlim\n\n/-- The telescoping tail: ∑' k, (1/(k+4) − 1/(k+5)) = 1/4. -/\nlemma tsum_one_div_four_sub_telescope :\n    (∑' k : ℕ, (1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5))) = 1 / 4 :=\n  hasSum_one_div_four_sub_telescope.tsum_eq\n\n/-- The telescope family is summable. -/\nlemma summable_one_div_four_sub_telescope :\n    Summable (fun k : ℕ => 1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5)) :=\n  hasSum_one_div_four_sub_telescope.summable\"\"\"\nassert old2 in src, \"old2 missing\"\nsrc = src.replace(old2, new2)\n\n# farW_shift5_summable: use the named summable\nold3 = \"\"\"  have htel : Summable (fun k : ℕ => 1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5)) :=\n    ⟨_, tsum_one_div_four_sub_telescope⟩\"\"\"\nnew3 = \"\"\"  have htel : Summable (fun k : ℕ => 1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5)) :=\n    summable_one_div_four_sub_telescope\"\"\"\nassert old3 in src, \"old3 missing\"\nsrc = src.replace(old3, new3)\n\n# farW_shift_summable: fix the ?_ structure\nold4 = \"\"\"/-- farW (k+1) is summable. -/\nlemma farW_shift_summable : Summable (fun k : ℕ => farW (k + 1)) := by\n  have h1 := (summable_nat_add_iff (f := fun k : ℕ => farW (k + 1)) 4).mp ?_\n  · exact h1\n  · apply farW_shift5_summable.congr\n    intro k\n    show farW (k + 4 + 1) = farW (k + 5)\n    rw [show k + 4 + 1 = k + 5 from rfl]\"\"\"\nnew4 = \"\"\"/-- farW (k+1) is summable. -/\nlemma farW_shift_summable : Summable (fun k : ℕ => farW (k + 1)) := by\n  apply (summable_nat_add_iff (f := fun k : ℕ => farW (k + 1)) 4).mp\n  apply farW_shift5_summable.congr\n  intro k\n  show farW (k + 4 + 1) = farW (k + 5)\n  rw [show k + 4 + 1 = k + 5 from rfl]\"\"\"\nassert old4 in src, \"old4 missing\"\nsrc = src.replace(old4, new4)\n\n# farW_exact_four: avoid norm_num on inequalities\nold5 = \"\"\"  have h1 : farW 1 ≤ 1 / 1.414 := by\n    rw [farW]\n    norm_num\n    have h2 : (1.414 : ℝ) ≤ Real.sqrt (1 + (1 : ℝ) ^ 2) := by\n      rw [show (1 : ℝ) + (1 : ℝ) ^ 2 = 2 from by ring]\n      exact hsqrt2\n    exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 1.414) h2\n  have h2' : farW 2 ≤ 1 / 4.46 := by\n    rw [farW]\n    have h3 : Real.sqrt (1 + (2 : ℝ) ^ 2) = Real.sqrt 5 := by\n      rw [show (1 : ℝ) + (2 : ℝ) ^ 2 = 5 from by ring]\n    rw [h3]\n    have h4 : (2 : ℝ) * 2.23 ≤ (2 : ℝ) * Real.sqrt 5 :=\n      mul_le_mul_of_nonneg_left hsqrt5 (by norm_num)\n    have h5 : (2 : ℝ) * 2.23 = 4.46 := by norm_num\n    rw [h5] at h4\n    exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 4.46) h4\n  have h3' : farW 3 ≤ 1 / 9.48 := by\n    rw [farW]\n    have h4 : Real.sqrt (1 + (3 : ℝ) ^ 2) = Real.sqrt 10 := by\n      rw [show (1 : ℝ) + (3 : ℝ) ^ 2 = 10 from by ring]\n    rw [h4]\n    have h5 : (3 : ℝ) * 3.16 ≤ (3 : ℝ) * Real.sqrt 10 :=\n      mul_le_mul_of_nonneg_left hsqrt10 (by norm_num)\n    have h6 : (3 : ℝ) * 3.16 = 9.48 := by norm_num\n    rw [h6] at h5\n    exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 9.48) h5\n  have h4' : farW 4 ≤ 1 / 16.492 := by\n    rw [farW]\n    have h5 : Real.sqrt (1 + (4 : ℝ) ^ 2) = Real.sqrt 17 := by\n      rw [show (1 : ℝ) + (4 : ℝ) ^ 2 = 17 from by ring]\n    rw [h5]\n    have h6 : (4 : ℝ) * 4.123 ≤ (4 : ℝ) * Real.sqrt 17 :=\n      mul_le_mul_of_nonneg_left hsqrt17 (by norm_num)\n    have h7 : (4 : ℝ) * 4.123 = 16.492 := by norm_num\n    rw [h7] at h6\n    exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 16.492) h6\n  linarith [h1, h2', h3', h4']\"\"\"\nnew5 = \"\"\"  have h1 : farW 1 ≤ 1 / 1.414 := by\n    have h2 : farW 1 = 1 / Real.sqrt 2 := by\n      rw [farW]\n      norm_num\n    rw [h2]\n    exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 1.414) hsqrt2\n  have h2' : farW 2 ≤ 1 / 4.46 := by\n    have h3 : farW 2 = 1 / ((2 : ℝ) * Real.sqrt 5) := by\n      rw [farW]\n      norm_num\n    rw [h3]\n    have h4 : (2 : ℝ) * 2.23 ≤ (2 : ℝ) * Real.sqrt 5 :=\n      mul_le_mul_of_nonneg_left hsqrt5 (by norm_num)\n    have h5 : (4.46 : ℝ) = 2 * 2.23 := by norm_num\n    rw [h5]\n    exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < (2 : ℝ) * 2.23) h4\n  have h3' : farW 3 ≤ 1 / 9.48 := by\n    have h4 : farW 3 = 1 / ((3 : ℝ) * Real.sqrt 10) := by\n      rw [farW]\n      norm_num\n    rw [h4]\n    have h5 : (3 : ℝ) * 3.16 ≤ (3 : ℝ) * Real.sqrt 10 :=\n      mul_le_mul_of_nonneg_left hsqrt10 (by norm_num)\n    have h6 : (9.48 : ℝ) = 3 * 3.16 := by norm_num\n    rw [h6]\n    exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < (3 : ℝ) * 3.16) h5\n  have h4' : farW 4 ≤ 1 / 16.492 := by\n    have h5 : farW 4 = 1 / ((4 : ℝ) * Real.sqrt 17) := by\n      rw [farW]\n      norm_num\n    rw [h5]\n    have h6 : (4 : ℝ) * 4.123 ≤ (4 : ℝ) * Real.sqrt 17 :=\n      mul_le_mul_of_nonneg_left hsqrt17 (by norm_num)\n    have h7 : (16.492 : ℝ) = 4 * 4.123 := by norm_num\n    rw [h7]\n    exact one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < (4 : ℝ) * 4.123) h6\n  linarith [h1, h2', h3', h4']\"\"\"\nassert old5 in src, \"old5 missing\"\nsrc = src.replace(old5, new5)\n\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\n"},"extra_content":null}]