[{"type":"thinking","thinking":"`Summable.hasSum_iff_tendsto_nat : Summable f → (HasSum f a ↔ Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 a))`.\n\nSo for the telescope: I can prove Summable FIRST (the family (1/(k+4) − 1/(k+5)) is summable as the difference of two \"shifted convergent\" families... hmm, or via the nonneg form: hasSum_iff_tendsto_nat_of_nonneg (hf : ∀ i, 0 ≤ f i) — MY family IS nonneg: 1/(k+4) − 1/(k+5) ≥ 0 ✓!! Use `hasSum_iff_tendsto_nat_of_nonneg (hf) (r) : HasSum f r ↔ Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 r)` — perfect: no summable needed, just nonneg + the range-tendsto.\n\nSo rewrite hasSum_one_div_four_sub_telescope:\n```\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 hnn : ∀ k : ℕ, 0 ≤ 1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5) := by\n    intro k\n    have h1 : ((k : ℝ) + 4) ≤ ((k : ℝ) + 5) := by linarith\n    have h2 : 1 / ((k : ℝ) + 5) ≤ 1 / ((k : ℝ) + 4) :=\n      one_div_le_one_div_of_le (by positivity : (0:ℝ) < (k:ℝ)+4) h1\n    linarith\n  rw [hasSum_iff_tendsto_nat_of_nonneg hnn]\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  ...\n```\nWait — hasSum_iff_tendsto_nat_of_nonneg is in ENNReal.lean?? It says {f : ℕ → ℝ} — OK it's for ℝ with nonneg. Good.\n\nFixes to apply:\n1. sum_congr: (fun i _ => h2 i).\n2. HasSum → hasSum_iff_tendsto_nat_of_nonneg route.\n3. farW_le_inv_mul_sub calls: (by omega : 2 ≤ k + 5).\n4. div_div fix for 1/((k+5)(k+4)) = 1/(k+4)/(k+5).\n5. Drop the bad third rw in farW_shift5_summable.\n6. farW_exact_four equalities with explicit rw chains.","signature":""},{"type":"text","text":"`hasSum_iff_tendsto_nat_of_nonneg` is the bridge (my family is nonneg). Applying all fixes."},{"type":"tool_use","id":"Write_83","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\patch_s10b.py","content":"p = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\n# 1. sum_congr per-element form\nold = \"\"\"    rw [Finset.sum_congr rfl h2] at h\"\"\"\nnew = \"\"\"    rw [Finset.sum_congr rfl (fun i _ => h2 i)] at h\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# 2. HasSum via the of_nonneg bridge\nold = \"\"\"  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\"\"\"\nnew = \"\"\"  have hnn : ∀ k : ℕ, (0 : ℝ) ≤ 1 / ((k : ℝ) + 4) - 1 / ((k : ℝ) + 5) := by\n    intro k\n    have h1 : 1 / ((k : ℝ) + 5) ≤ 1 / ((k : ℝ) + 4) :=\n      one_div_le_one_div_of_le (by positivity : (0 : ℝ) < (k : ℝ) + 4) (by linarith)\n    linarith\n  rw [hasSum_iff_tendsto_nat_of_nonneg hnn]\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\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# 3. farW_le_inv_mul_sub hypothesis at call sites (k+5: 2 ≤ k+5)\nsrc = src.replace(\"farW_le_inv_mul_sub (k := k + 5) (by omega : 1 ≤ k + 5)\",\n                  \"farW_le_inv_mul_sub (k := k + 5) (by omega : 2 ≤ k + 5)\")\n\n# 4. div form: 1/((k+5)*(k+4)) = 1/(k+4)/(k+5) via div_div + mul_comm\nold = \"\"\"      _ = 1 / ((k : ℝ) + 4) / ((k : ℝ) + 5) := by\n          rw [div_mul_div_comm, mul_comm ((k : ℝ) + 5) ((k : ℝ) + 4), one_mul]\"\"\"\nnew = \"\"\"      _ = 1 / ((k : ℝ) + 4) / ((k : ℝ) + 5) := by\n          rw [div_div, mul_comm ((k : ℝ) + 5) ((k : ℝ) + 4)]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# 5. drop the bad third rw in farW_shift5_summable\nold = \"\"\"  rw [Real.norm_eq_abs, abs_of_nonneg (farW_nonneg _),\n    abs_of_nonneg (by positivity : (0 : ℝ) ≤ 1 / ((k : ℝ) + 4) / ((k : ℝ) + 5))]\n  exact h2\"\"\"\nnew = \"\"\"  rw [Real.norm_eq_abs, abs_of_nonneg (farW_nonneg _)]\n  exact h2\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# 6. farW_exact_four equalities via explicit rw chains\nold = \"\"\"  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]\"\"\"\nnew = \"\"\"  have h1 : farW 1 ≤ 1 / 1.414 := by\n    have h2 : farW 1 = 1 / Real.sqrt 2 := by\n      rw [farW, Nat.cast_one, one_pow, show (1 : ℝ) + 1 = 2 from rfl, one_mul]\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, Nat.cast_ofNat, show (2 : ℝ) ^ 2 = 4 by norm_num,\n        show (1 : ℝ) + 4 = 5 by norm_num]\n    rw [h3]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\nold = \"\"\"  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]\"\"\"\nnew = \"\"\"  have h3' : farW 3 ≤ 1 / 9.48 := by\n    have h4 : farW 3 = 1 / ((3 : ℝ) * Real.sqrt 10) := by\n      rw [farW, Nat.cast_ofNat, show (3 : ℝ) ^ 2 = 9 by norm_num,\n        show (1 : ℝ) + 9 = 10 by norm_num]\n    rw [h4]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\nold = \"\"\"  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]\"\"\"\nnew = \"\"\"  have h4' : farW 4 ≤ 1 / 16.492 := by\n    have h5 : farW 4 = 1 / ((4 : ℝ) * Real.sqrt 17) := by\n      rw [farW, Nat.cast_ofNat, show (4 : ℝ) ^ 2 = 16 by norm_num,\n        show (1 : ℝ) + 16 = 17 by norm_num]\n    rw [h5]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\n"},"extra_content":null}]