[{"type":"thinking","thinking":"1. `[[↑k, ↑k + 1]]` — the uIcc notation needs `open Set` or the `Set.uIcc` — the `[[ ]]` notation requires `open Set` (scoped). In LiContour we had `open ... Topology` — uIcc notation is in `Set` scope. Options: add `open Set`? risky (name clashes). Better: use `Set.uIcc (k:ℝ) (k+1:ℝ)` explicitly.\n\n2. hIBP consequence (line 184): after rw [hb1, hb2] + simp, hIBP : ∫ u*v' = 0 − ∫ u'*v. My h2 wanted: ∫ u'*v = −∫ u*v' via linarith — ℂ equation!! linarith can't do ℂ. Fix: hIBP : A = 0 − B ⟹ B = −A: `eq_neg_of_eq...`? From hIBP : ∫u*v' = 0 − ∫u'*v: then ∫u'*v = −∫u*v': `eq_neg_of_sub_eq`? hIBP : a = 0 − b ⟹ b = −a: `by linarith` no — use: have h2 : B = −A := by rw [hIBP]; ring? hIBP : A = 0 − B: goal B = −A: rw [hIBP]: B = −(0 − B): ring? — for ℂ: ring works ✓: `have h2 : (∫ ...) = -(∫ u*v') := by rw [hIBP]; ring`. Let me restructure the ending:\n   hIBP : ∫ u*v' = 0 − ∫ u'*v.\n   Goal: ∫ u'*v = (s+1)·∫ u·x^(−s−2) hmm wait the goal is ∫ B1·x^(−s−1) = (s+1)·∫ b2Fun·x^(−s−2).\n   From hIBP: ∫ b2Fun·(−(s+1))x^{−s−2} = −∫ B1·x^{−s−1}.\n   LHS = −(s+1)∫ b2Fun·x^{−s−2} (const_mul pullout).\n   So: −(s+1)∫b2Fun·.. = −∫B̃₁·.. ⟹ ∫B̃₁·.. = (s+1)∫b2Fun·.. ✓.\n   ```lean\n   rw [show (fun x : ℝ => (b2Fun x : ℂ) * ((-(s + 1)) * (x : ℂ) ^ (-s - 2)))\n       = fun x : ℝ => (-(s + 1)) * ((b2Fun x : ℂ) * (x : ℂ) ^ (-s - 2)) from\n     funext fun x => by ring] at hIBP\n   rw [intervalIntegral.integral_const_mul] at hIBP\n   -- hIBP : (−(s+1)) * ∫b2Fun·.. = −∫ B̃₁·..\n   have h2 : ∫ x in (k:ℝ)..(k+1:ℝ), (B1 x : ℂ) * (x:ℂ)^(-s-1) = (s+1) * ∫ x in (k:ℝ)..(k+1:ℝ), (b2Fun x:ℂ)*(x:ℂ)^(-s-2) := by\n     rw [hIBP]  -- hmm direction\n     ring\n   exact h2\n   ```\n   From hIBP : −(s+1)·B = −A (A = ∫B̃₁.., B = ∫b2..): goal A = (s+1)B: rw?? Let me: have h2 : ... := by linear... ℂ again. Use: `neg_eq_neg`..: from hIBP: A = (s+1)·B: `eq_neg_iff...`: simplest: \n   ```lean\n   have h2 : ∫ ... B1 ... = (s + 1) * ∫ ... b2Fun ... := by\n     have h := hIBP  -- (−(s+1)) * B = 0 − A\n     rw [sub_zero] at h  -- h : (−(s+1)) * B = −A — wait hIBP after the earlier simp [hb1,hb2]: ∫u*v' = 0 − ∫u'*v: after my rewrites hIBP : (−(s+1))*B = 0 − A: rw [zero_sub] at hIBP: (−(s+1))*B = −A: then A = (s+1)*B: neg_injective: `neg_inj.mp`? from −X = −Y ⟹ X = Y: neg_inj : −a = −b ↔ a = b ✓: hIBP : (−(s+1))*B = −A — rewrite LHS −(s+1)*B = −((s+1)*B) (neg_mul ✓): then neg_inj.mp : (s+1)*B = A: .symm ✓.\n   have h3 : ∫ ... B1 ... = (s+1) * ∫ ... := (neg_inj.mp (by rw [neg_mul] at hIBP... )).symm\n   ```\n   Concrete:\n   ```lean\n   rw [intervalIntegral.integral_const_mul, zero_sub] at hIBP\n   -- hIBP : -(s + 1) * J = -I  where J = ∫b2.., I = ∫B1..\n   rw [neg_mul] at hIBP  -- −((s+1)*J) = −I\n   exact (neg_inj.mp hIBP).symm\n   ```\n   And drop my old h2/rw-chain. The final `exact (neg_inj.mp hIBP).symm` — hIBP : −((s+1)*J) = −I: neg_inj.mp : (s+1)*J = I: .symm : I = (s+1)*J ✓ goal.\n\n3. uIcc notation → Set.uIcc explicit.\n\nLet me rewrite the tail of the proof.","signature":""},{"type":"text","text":"Fixing the uIcc notation (needs `Set.` prefix) and the final ℂ-algebra (linarith can't do ℂ)."},{"type":"tool_use","id":"Edit_221","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)) [[(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","new_string":"  have hu : ContinuousOn (fun x : ℝ => (b2Fun x : ℂ)) (Set.uIcc (k : ℝ) (k + 1 : ℝ)) :=\n    (continuous_ofReal.comp continuous_b2Fun).continuousOn\n  have hv : ContinuousOn (fun x : ℝ => (x : ℂ) ^ (-s - 1)) (Set.uIcc (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}]