[{"type":"thinking","thinking":"Errors:\n1. Line 161/162: in hv'int — `ContinuousOn.intervalIntegrable` — my application `apply ContinuousOn.intervalIntegrable` then `intro x hx` gave hx : x ∈ Set.uIcc (not Icc) — ContinuousOn.intervalIntegrable's hypothesis form: `ContinuousOn.intervalIntegrable (h : ContinuousOn f (uIcc a b)) : IntervalIntegrable f volume a b`? Actually: `ContinuousOn.intervalIntegrable (hf : ContinuousOn f [[a,b]])` — the goal after `apply` is ContinuousOn f (uIcc a b)? and intro x hx gives uIcc membership. hx.1 : min ↑k (↑k+1) ≤ x — linarith [hx.1] fails since hx.1 has min. Fix: `have hx0 : x ≠ 0 := by have := hx.1; rw [min_eq_left (by linarith : (k:ℝ) ≤ k+1)] at this; linarith` or simpler: membership in uIcc k (k+1) with k ≤ k+1 gives Icc: `rw [Set.uIcc_of_le (by linarith : (k:ℝ) ≤ k+1)] at hx` first.\n\n2. hr2's proof: `fun h => hs (by linarith [neg_eq_zero.mp ...])` — h : -s - 2 = 0 is ℂ — linarith fails on ℂ again! My `(by linarith : s = -2 - 0)` is a ℂ goal — linarith can't. Fix ℂ-side: from h : -s - 2 = 0: s = -2: sub_eq_zero.mp h : -s = 2: then s = -2: neg_eq_iff_eq_neg... : -s = 2 ⟹ s = -2 ✓ then hs needs s = -1: from s = -2... wait hs : s ≠ -1 — from s = -2 I can't conclude s = -1!! LOGIC ERROR: hr2 : (-s - 2) ≠ 0 — from s ≠ −1?? That's WRONG: −s−2 = 0 ⟺ s = −2 — NOT excluded by s ≠ −1!! I need to exclude s = −2 for the SECOND cpow's derivative... wait — hasDerivAt_ofReal_cpow_const (hx : x ≠ 0) (hr : r ≠ 0) — the hr requirement r ≠ 0 is for the derivative FORMULA r*x^(r−1) — for r = 0: x^0 = 1 constant, derivative 0, formula gives 0·x^{−1} = 0 ✓ the formula is actually still right but the lemma requires r ≠ 0. For r = 0 case: hasDerivAt_const. Hmm — but do I even need the x^(−s−2) HasDerivAt separately? For hv'int (integrability of v') I only need CONTINUITY of x^(−s−2), not the derivative! Continuity of (x:ℂ)^(−s−2) at x ≠ 0 for ANY exponent: `continuousAt_ofReal_cpow_const (x) (y) (h : 0 < y.re ∨ x ≠ 0)` — from the grep: `continuousAt_ofReal_cpow_const (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0) : ContinuousAt (fun x => (↑x)^y) x` ✓ — no exponent-ne-zero needed! Use that for hv and hv'int:\n```lean\n    exact (continuousAt_ofReal_cpow_const x _ (Or.inr hx0)).continuousWithinAt\n```\nMuch cleaner — no hr/hr2 needed for continuity. (Keep hr for the derivative value in hvv'.)\n\n3. Line 168: the IBP application — the lemma's interval forms use uIcc not Icc: my hu/hv are on Set.Icc but the lemma wants ContinuousOn on [[a,b]] = uIcc. Fix: hu/hv should be on Set.uIcc — or provide Icc and convert: uIcc k (k+1) = Icc k (k+1) since k ≤ k+1: `Set.uIcc_of_le`. Change hu/hv to state on [[k, k+1]] via uIcc: write the ContinuousOn over `Set.uIcc (k:ℝ) (k+1:ℝ)` directly, or after proving on Icc, convert: `(Set.uIcc_of_le (by linarith : (k:ℝ) ≤ k+1)) ▸ hu`. Simplest: state hu/hv with uIcc and in their proofs rewrite uIcc → Icc at the start:\n```lean\n  have hu : ContinuousOn (fun x : ℝ => (b2Fun x : ℂ)) [[k, k+1]] :=\n    (continuous_ofReal.comp continuous_b2Fun).continuousOn  -- ContinuousOn on univ ⊇ anything: continuousOn is on the WHOLE space — fine for uIcc too? continuousOn : ContinuousOn f univ? no — .continuousOn : ContinuousOn f s for ANY s? Continuous.continuousOn : ContinuousOn f s for any s ✓ (Continuous.continuousOn (hf : Continuous f) : ContinuousOn f s) — yes any s ✓ fine as-is.\n```\nFor hv: prove on uIcc:\n```lean\n  have hv : ContinuousOn (fun x : ℝ => (x : ℂ) ^ (-s - 1)) [[(k : ℝ), (k + 1 : ℝ)]] := by\n    rw [Set.uIcc_of_le (by linarith : (k:ℝ) ≤ k + 1)]\n    intro x hx\n    have hx0 : x ≠ 0 := by linarith [hx.1]\n    exact (continuousAt_ofReal_cpow_const x _ (Or.inr hx0)).continuousWithinAt\n```\nSimilarly hv'int's function: (−(s+1))·x^(−s−2): const_mul of the same with exponent −s−2: IntervalIntegrable.const_mul (IntervalIntegrable of x^(−s−2)): the x^(−s−2) integrable via ContinuousOn.intervalIntegrable (uIcc form) + continuousAt_ofReal_cpow_const ✓ (no hr2 needed).\n\n4. Line 183 linarith fail — cascade probably. Let me apply fixes.","signature":""},{"type":"text","text":"Fixing: uIcc vs Icc forms, and using `continuousAt_ofReal_cpow_const` (no exponent condition needed) for continuity."},{"type":"tool_use","id":"Edit_218","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean","old_string":"  have hu : ContinuousOn (fun x : ℝ => (b2Fun x : ℂ)) (Set.Icc (k : ℝ) (k + 1 : ℝ)) :=\n    (continuous_ofReal.comp continuous_b2Fun).continuousOn\n  have hv : ContinuousOn (fun x : ℝ => (x : ℂ) ^ (-s - 1)) (Set.Icc (k : ℝ) (k + 1 : ℝ)) := by\n    intro x hx\n    have hx0 : x ≠ 0 := by linarith [hx.1]\n    exact (hasDerivAt_ofReal_cpow_const hx0 hr).continuousAt.continuousWithinAt","new_string":"  have hu : ContinuousOn (fun x : ℝ => (b2Fun x : ℂ)) [[(k : ℝ), (k + 1 : ℝ)]] :=\n    (continuous_ofReal.comp continuous_b2Fun).continuousOn\n  have hv : ContinuousOn (fun x : ℝ => (x : ℂ) ^ (-s - 1)) [[(k : ℝ), (k + 1 : ℝ)]] := by\n    rw [Set.uIcc_of_le (by linarith : (k:ℝ) ≤ k + 1)]\n    intro x hx\n    have hx0 : x ≠ 0 := by linarith [hx.1]\n    exact (continuousAt_ofReal_cpow_const x _ (Or.inr hx0)).continuousWithinAt"},"extra_content":null}]