[{"type":"thinking","thinking":"Several small fixes:\n\n1. **Line 405** (the first calc step): `ring` after simp only — \"No goals\" — simp closed it: remove `ring`.\n\n2. **Line 419**: the final bound-equality: `x^(−(σ+3)/2)·|log x| = |x|^(−(σ+3)/2)·|log x|` — Real.abs_rpow_of_nonneg rewrote |x^(...)| into |x|^(...)!! The lemma: Real.abs_rpow_of_nonneg (hx : 0 ≤ x) : |x^y| = |x|^y? hmm — it rewrote |x^y| → |x|^y — then need |x| = x: add `abs_of_pos (by linarith)`:\n```lean\n      _ = bound x := by\n            simp only [bound, mul_one]\n            rw [Real.abs_rpow_of_nonneg (by linarith), abs_of_pos (by linarith : (0:ℝ) < x)]\n```\nHmm — Real.abs_rpow_of_nonneg (hx : 0 ≤ x) : |x ^ y| = |x| ^ y ✓ then abs_of_pos : |x| = x ✓.\n\n3. **Line 446/447**: in h_diff's final convert block:\n```lean\n    convert h2 using 1\n    · ext w\n      simp only [F]\n      ring\n    · simp only [F', f]\n      rw [Complex.ofReal_log (by linarith : (0:ℝ) ≤ x)]\n      push_cast\n      ring\n```\nThe second bullet goal: F' z x = ↑x^(−z−2)·log ↑x·(−1)·f x — my `rw [Complex.ofReal_log]` — but the goal has `log ↑x` (Complex.log of ↑x) — ofReal_log : log ↑r = ↑(Real.log r) — rw rewrites log ↑x → ↑(Real.log x) ✓ then push_cast; ring. Error 447: \"simp made no progress\" at 447:6 — hmm 446 \"No goals\" — the bullets mis-ordered: convert h2 using 1 produced goals in different order (function-eq first or value-eq first?). The errors: 446:6 \"No goals\" and 447:6 \"simp made no progress\" — these are the two bullet tactics. And 433:89 unsolved: `F' z x = ↑x^(−z−2)·log↑x·(−1)·f x` — the second goal unsolved... wait the errors are confusing. The 433:89 error (unsolved, case e'_9) shows goal F' z x = ...·f x — that's the VALUE-matching goal of the convert — but actually it's the FUNCTION?? convert h2 using 1: h2 : HasDerivAt (fun y => ↑x^(−y−2)·f x) (...) z; goal: HasDerivAt (fun w => F w x) (F' z x) z — using 1: goals F' z x = (the derivative) AND (fun w => F w x) = (fun y => ↑x^(−y−2)·f x) — order unknown. The unsolved goal `F' z x = ↑x^(−z−2)·log↑x·(−1)·f x` remained — my second bullet was supposed to close it: `simp only [F', f]` then rw ofReal_log, push_cast, ring — the simp made no progress (447) — because after convert, F' z x was already unfolded?? If convert's congruence already unfolded F'/f (they're set-bound let vars...). The remaining goal displayed `F' z x = ...` WITH F' still folded — so simp only [F', f] SHOULD unfold... \"made no progress\" — hmm — because F' is a let-variable introduced by `set` — simp only [F'] unfolds a set variable? `set` introduces F' as a local def; simp only [F'] works on the hypothesis name... Actually `simp only [F']` where F' is a let-bound local — that works if F' is a local hypothesis with an equation... set creates `F' : ℂ → ℝ → ℂ := fun z x => ...` and a hypothesis? No — set replaces occurrences and the unfolding is via `simp only [F']`?? For set-variables, unfolding uses `simp only [hF'_def]`?? No — set doesn't create an hF'_def by default... Actually `set x := e with h` creates h : x = e. Without `with`, no equation hypothesis — the unfolding of F' in goals is via `show` or `simp only [F']`?? Hmm — earlier `simp only [F', f]` DID work (in hF_meas etc. it compiled). So simp only [F'] does unfold set-vars — OK. Then why \"made no progress\" at 447?\n\nLooking at the error sequence more carefully: 446:6 \"No goals to be solved\" — that's my `rw [Complex.ofReal_log ...]`?? and 447:6 simp no progress — the bullets: `· ext w; simp only [F]; ring` (446?) and `· simp only [F', f]; rw...` — I think convert produced ONE goal only (the ext/... — hmm.\n\nAnd 433:89 (case e'_9 unsolved) — `F' z x = ↑x^(−z−2)·log↑x·(−1)·f x` — position 433:89 — that's the `convert h2 using 1` line? Line 433 is `convert h2 using 1`... The goals from convert: (a) the value equation (F' z x = ↑x^·...) — shown unsolved, meaning my bullets didn't fire on it correctly.\n\nThe problem might be: convert h2 using 1 produces goals: first the FUNCTION equality (fun w ↦ F w x = fun y ↦ ↑x^(−y−2)·f x), second the value (F' z x = derivative). My first bullet `ext w; simp only [F]; ring`: ext w on function equality ✓ then simp only [F] unfolds F — goal F w x = ↑x^(−w−2)·f x → after simp only [F]: (↑x^(−w−2)·f x) = (↑x^(−w−2)·f x)?? — closes by rfl... then `ring` — no goals → error 446 \"No goals\". So simp only [F] closed it (rfl after unfold). Remove ring from first bullet.\nSecond bullet: `simp only [F', f]` — \"made no progress\" — because convert... hmm the second goal might have F' ALREADY unfolded by convert (congruence unfolds set-vars?). The unsolved goal display shows `F' z x = ...` — hmm it shows F' NOT unfolded. And simp only [F', f] \"made no progress\"?? Weird — maybe because simp only [F'] on a set-variable inside a convert-generated goal... \n\nHonestly, cleaner: avoid convert; use explicit equality lemmas:\n```lean\n    have hfun : (fun w ↦ F w x) = (fun y ↦ (x : ℂ) ^ (-y - 2) * f x) := by\n      ext w\n      simp only [F]\n    have hval : F' z x = (x : ℂ) ^ (-z - 2) * Complex.log (x : ℂ) * (-1) * f x := by\n      simp only [F', f]\n      rw [Complex.ofReal_log (by linarith : (0:ℝ) ≤ x)]\n      push_cast\n      ring\n    rw [hfun, hval]\n    exact h2\n```\nHmm — but rw [hfun, hval] on the goal — goal is HasDerivAt (fun z => ∫...)... no wait — h_diff's per-x goal: HasDerivAt (fun w ↦ F w x) (F' z x) z — rw [hfun, hval] rewrites the function and value ✓ then exact h2 ✓. But the FIRST bullet's simp only [F] might again auto-close... `ext w; simp only [F]` — the goal F w x = ↑x^(−w−2)·f x — after simp only [F]: (↑x^(−w−2)·f x) = ↑x^(−w−2)·f x — closed by rfl automatically as part of simp ✓ no error (simp closing the goal is fine — no following tactic). ✓.\n\nFor hval: simp only [F', f] unfolds; goal: ↑x^(−z−2)·(−↑(Real.log x))·↑(b2Fun x) = ↑x^(−z−2)·log↑x·(−1)·↑(b2Fun x): rw [Complex.ofReal_log]: log↑x → ↑(Real.log x): then push_cast; ring ✓ — but will `simp only [F', f]` + rw fire cleanly... The `−↑(Real.log x)` vs `−(↑(Real.log x))`... push_cast normalizes ✓ ring ✓.\n\n4. **Line 456**: the final convert's first bullet: `ext z; apply integral_congr_ae` — error: integral_congr_ae unification with goal `z + x✝ = z + x✝`?! — the ext z produced a junk goal — because the goal after convert key using 1 wasn't the function equation I expected. The `key` : HasDerivAt (fun n ↦ ∫ a, F n a ∂μ) (∫ a, F' s a ∂μ) s and my goal: HasDerivAt (fun z => ∫ x in Ioi 1, ↑b2Fun x·↑x^(−z−2)) (∫ x in Ioi 1, ↑b2Fun x·(↑x^(−s−2)·(−↑log x))) s. convert key using 1: the function equation: (fun z => ∫ x in Ioi 1, mine) = (fun n => ∫ a, F n a ∂μ): ext z: goal ∫ x in Ioi 1, mine = ∫ a, F z a ∂μ: then I did `apply integral_congr_ae` — but the error shows a goal `z + x✝ = z + x✝` — nonsense — because ext z already... hmm — \"ext z\" applied to function-equality gives the pointwise goal ✓; then apply integral_congr_ae — the goal is an equality of INTEGRALS with DIFFERENT measures (∂volume-on-Ioi vs ∂μ-restrict) — integral_congr_ae expects same measure! The error's weird goal suggests the unification went sideways.\n\nThe set-integral vs restrict-integral: ∫ x in Ioi 1, g x ∂volume = ∫ x, g x ∂(volume.restrict (Ioi 1)) — definitional (set integral IS the restrict integral): `MeasureTheory.integral_restrict`? Actually `∫ x in s, f x ∂μ` is NOTATION/def for ∫ x, f x ∂(μ.restrict s) — so they're defeq — but convert's congruence might not see through. Better approach: state `key`'s conclusion directly in Ioi-form first:\n```lean\n  have key2 : HasDerivAt (fun z : ℂ => ∫ x in Set.Ioi (1:ℝ), (b2Fun x : ℂ) * (x : ℂ) ^ (-z - 2))\n      (∫ x in Set.Ioi (1:ℝ), (b2Fun x : ℂ) * ((x : ℂ) ^ (-s - 2) * (-Real.log x))) s := by\n    convert key using 1\n    · -- function eq\n      ext z\n      rw [show (∫ a, F z a ∂μ) = ∫ x in Set.Ioi (1:ℝ), F z x from rfl]  -- set-integral defeq?\n      apply integral_congr_ae\n      filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with x hx\n      simp only [F, f]\n      ring\n    · -- value eq\n      rw [show (∫ a, F' s a ∂μ) = ∫ x in Set.Ioi (1:ℝ), F' s x from rfl]\n      apply integral_congr_ae\n      filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with x hx\n      simp only [F', f]\n      push_cast\n      ring\n  exact key2\n```\nHmm — the `from rfl` for the set/restrict identity: `∫ a, F z a ∂μ` with μ = volume.restrict (Ioi 1) — μ is a let (let μ : Measure ℝ := ...) — `∫ x in Ioi 1, F z x ∂volume` unfolds to `∫ a, F z a ∂(volume.restrict (Ioi 1))` — the let-var μ needs unfolding — `rfl` might work via defeq... risky. Alternative cleaner: skip `let μ` entirely — state everything with (volume.restrict (Set.Ioi 1)) directly? The Zeta23 proof used `let μ`. Hmm. \n\nAlternative simplest: avoid the final convert entirely by stating the theorem in the ∂-form! I.e., change hasDerivAt_b2Integral's statement to use ∫ ... ∂(volume.restrict ...)?? No — I want the Ioi form for downstream.\n\nBetter: use `MeasureTheory.integral_restrict`? hmm — the identity `∫ x in s, f x ∂μ = ∫ x, f x ∂(μ.restrict s)` is `MeasureTheory.integral_restrict`?? I believe the def is the other way: set-integral is DEFINED as the restrict integral, so `∫ x in s, f x ∂μ ≡ ∫ x, f x ∂(μ.restrict s)` by rfl. So in convert, the functions differ by this defeq — convert using 1 might just handle it via defeq (congruence descends into the measure argument? convert uses congruence lemmas; the measure position is an argument of the integral constant — congr should handle via defeq when it can't match syntactically... risky).\n\nCleanest robust: define μ NOT via let but inline the restrict everywhere, and end the theorem statement in the ∂-form, then a separate trivial corollary rewrites to Ioi form via rfl... Let me just try `exact key` variants at compile — actually simplest experiment: keep the convert but replace the bullets' integral_congr_ae with a `rfl`-attempt first... \n\nActually — the real subtlety: after `convert key using 1`, the function-equality goal: (fun z => ∫ x in Ioi 1, ↑b2Fun x * ↑x^(−z−2) ∂volume) = (fun n => ∫ a, F n a ∂μ). Since ∫-in-set is defeq to ∫-restrict: the LHS function = fun z => ∫ a, (↑b2Fun a·↑a^(−z−2)) ∂(restrict) and RHS = fun z => ∫ a, (↑a^(−z−2)·↑b2Fun a) ∂(restrict) — equal via integral_congr_ae + mul_comm. So: ext z; apply integral_congr_ae; filter_upwards; simp only [F, f]; ring — but `apply integral_congr_ae` on goal `∫ x in Ioi 1, ... ∂volume = ∫ a, F z a ∂μ` — the measures differ syntactically (volume vs μ-restrict) BUT after `ext z`, the goal might still display in the set-integral notation — and integral_congr_ae's unification of measures: μ vs volume.restrict (Ioi 1) — μ is a let-var — unfolding... The error showed a junk goal `z + x✝ = z + x✝` — suggests ext z went wrong (ext on a non-function-equality?). Maybe convert using 1 already closed the function part by defeq and left only the value part — so my `ext z` hit the VALUE goal (an integral equality, not a function equality — ext z there produces junk).\n\nNew approach: skip convert; build the final result directly from key with `EventuallyEq`-free rewrites:\n```lean\n  have keyI : HasDerivAt (fun z : ℂ => ∫ x in Set.Ioi (1:ℝ), F z x)\n      (∫ x in Set.Ioi (1:ℝ), F' s x) s := key  -- defeq of set-integral and restrict-integral\n```\nThis `key`'s type has ∂μ with μ := let-bound restrict — the ASCRIPTION to the Ioi form forces the defeq check — should work (set integral unfolds to restrict integral, μ unfolds) — `exact key` might even directly work for the ascription. Then:\n```lean\n  convert keyI using 1\n  · ext z; apply integral_congr_ae; filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with x hx\n    simp only [F, f]; ring\n  · apply integral_congr_ae; filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with x hx\n    simp only [F', f]; push_cast; ring\n```\nHere keyI's integrals are genuine set-integrals (Ioi form) so integral_congr_ae unifies fine. The goal (my theorem statement) vs keyI: function: fun z => ∫ Ioi, ↑b2·↑x^(−z−2) vs fun z => ∫ Ioi, F z x = ↑x^(−z−2)·↑b2: per-point ring ✓; value similar ✓.\n\nLet me apply all these fixes.","signature":""},{"type":"text","text":"Restructuring the ending: direct ascription for the set/restrict integral defeq, and explicit equality lemmas in h_diff."},{"type":"tool_use","id":"Edit_265","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean","old_string":"    calc ‖F' z x‖ = ‖(x : ℂ) ^ (-z - 2)‖ * |Real.log x| * |b2Fun x| := by\n            simp only [F', f, norm_mul, norm_neg, Complex.norm_real, Real.norm_eq_abs]\n            ring\n      _ ≤ x ^ (-(s.re + 3) / 2) * |Real.log x| * (1 / 8) := by\n            have h1 : ‖(x : ℂ) ^ (-z - 2)‖ * |Real.log x|\n                ≤ x ^ (-(s.re + 3) / 2) * |Real.log x| :=\n              mul_le_mul_of_nonneg_right hpow (abs_nonneg _)\n            calc ‖(x : ℂ) ^ (-z - 2)‖ * |Real.log x| * |b2Fun x|\n                ≤ (x ^ (-(s.re + 3) / 2) * |Real.log x|) * |b2Fun x| :=\n                  mul_le_mul_of_nonneg_right h1 (abs_nonneg _)\n              _ ≤ (x ^ (-(s.re + 3) / 2) * |Real.log x|) * (1 / 8) :=\n                  mul_le_mul_of_nonneg_left (abs_b2Fun_le x)\n                    (mul_nonneg (Real.rpow_nonneg (by linarith) _) (abs_nonneg _))\n      _ ≤ x ^ (-(s.re + 3) / 2) * |Real.log x| * 1 :=\n            mul_le_mul_of_nonneg_left (by norm_num : (1/8 : ℝ) ≤ 1)\n              (mul_nonneg (Real.rpow_nonneg (by linarith) _) (abs_nonneg _))\n      _ = bound x := by\n            simp only [bound, mul_one]\n            rw [Real.abs_rpow_of_nonneg (by linarith)]","new_string":"    calc ‖F' z x‖ = ‖(x : ℂ) ^ (-z - 2)‖ * |Real.log x| * |b2Fun x| := by\n            simp only [F', f, norm_mul, norm_neg, Complex.norm_real, Real.norm_eq_abs]\n            ring\n      _ ≤ x ^ (-(s.re + 3) / 2) * |Real.log x| * (1 / 8) := by\n            have h1 : ‖(x : ℂ) ^ (-z - 2)‖ * |Real.log x|\n                ≤ x ^ (-(s.re + 3) / 2) * |Real.log x| :=\n              mul_le_mul_of_nonneg_right hpow (abs_nonneg _)\n            calc ‖(x : ℂ) ^ (-z - 2)‖ * |Real.log x| * |b2Fun x|\n                ≤ (x ^ (-(s.re + 3) / 2) * |Real.log x|) * |b2Fun x| :=\n                  mul_le_mul_of_nonneg_right h1 (abs_nonneg _)\n              _ ≤ (x ^ (-(s.re + 3) / 2) * |Real.log x|) * (1 / 8) :=\n                  mul_le_mul_of_nonneg_left (abs_b2Fun_le x)\n                    (mul_nonneg (Real.rpow_nonneg (by linarith) _) (abs_nonneg _))\n      _ ≤ x ^ (-(s.re + 3) / 2) * |Real.log x| * 1 :=\n            mul_le_mul_of_nonneg_left (by norm_num : (1/8 : ℝ) ≤ 1)\n              (mul_nonneg (Real.rpow_nonneg (by linarith) _) (abs_nonneg _))\n      _ = bound x := by\n            simp only [bound, mul_one]\n            rw [Real.abs_rpow_of_nonneg (by linarith), abs_of_pos (by linarith : (0:ℝ) < x)]"},"extra_content":null}]