[{"type":"thinking","thinking":"Wait — I left a broken §4 with a sorry and a malformed filter_upwards (the third filter `eventually_ne_nhds (show (1:ℂ) ≠ 1 ...)` is nonsense — eventually_ne_nhds (h : a ≠ b) : ∀ᶠ x in 𝓝 a, x ≠ b; I want ∀ᶠ s in 𝓝[≠] 1, s ≠ 1 which is self_mem_nhdsWithin — already the first). Let me fix §4 properly before compiling:\n\nlogDeriv_zeta_eq_logDeriv_zetaA_sub: for s ≠ 1 near 1 with zetaA s ≠ 0:\n- ζ s ≠ 0: zetaA s = (s−1)ζ s (zetaA_apply hs1) ⟹ ζ s = zetaA s/(s−1) ≠ 0 ✓.\n- zetaA =ᶠ[𝓝 s] (fun w => (w−1) * riemannZeta w): eventually_ne_nhds hs1 gives ∀ᶠ z in 𝓝 s, z ≠ 1; then zetaA_apply ✓.\n- logDeriv zetaA s = logDeriv ((w−1)·ζ w) s (congr via the =ᶠ: logDeriv_apply + deriv_eq + self_of_nhds) = logDeriv (w−1) s + logDeriv ζ s (logDeriv_mul: (s−1) ≠ 0, ζ s ≠ 0, DifferentiableAt (w−1) ✓, ζ differentiable at s ≠ 1 ✓ (differentiableAt_riemannZeta hs1 — check name: analyticAt_riemannZeta hs1 : AnalyticAt; .differentiableAt ✓)) = 1/(s−1) + logDeriv ζ s (logDeriv (w−1): deriv (w−1) = 1: logDeriv_apply: deriv f s/f s: deriv (fun w => w−1) s = 1: (differentiableAt_id.sub (differentiableAt_const 1)).deriv simp: deriv sub = deriv id − deriv const = 1 − 0 = 1: HasDerivAt.deriv of (hasDerivAt_id s).sub_const... simplest: `simp [logDeriv_apply]` with deriv of (w−1): there's `deriv_sub_const : deriv (fun x => x − c) x = 1`? `deriv_sub_const : deriv (fun x => x - c) x = 1` ✓ exists I believe (deriv_id' sub const). Let me just compute: logDeriv (fun w => w − 1) s = deriv (fun w => w − 1) s / (s − 1) = 1/(s−1): have : deriv (fun w : ℂ => w - 1) s = 1 := by simp [deriv_sub_const?]... `deriv_sub_const : deriv (fun x => x - c) = fun x => 1`?? form: `deriv_sub_const (c) : deriv (fun x => x - c) x = 1`? I'll write `by simp` and hope; fallback `(hasDerivAt_id' s).sub_const 1 |>.deriv` hmm hasDerivAt_id' : HasDerivAt id 1 x... (hasDerivAt_id x).sub_const? `HasDerivAt.sub_const : HasDerivAt f f' x → HasDerivAt (fun x => f x - c) f' x` ✓ then .deriv : deriv = 1 ✓. I'll use that, robust.\n\nSo:\n```lean\nlemma logDeriv_zeta_eq_logDeriv_zetaA_sub :\n    ∀ᶠ s in 𝓝[≠] (1 : ℂ),\n      logDeriv riemannZeta s = logDeriv zetaA s - 1 / (s - 1) := by\n  filter_upwards [self_mem_nhdsWithin,\n    mem_nhdsWithin_of_mem_nhds zetaA_ne_zero_near_one] with s hs1 hZA\n  have hs1' : s - 1 ≠ 0 := sub_ne_zero.mpr hs1\n  have hζ : riemannZeta s ≠ 0 := by\n    have e := zetaA_apply hs1\n    intro hz\n    rw [hz, mul_zero] at e\n    exact hZA e.symm\n  have hcong : zetaA =ᶠ[𝓝 s] (fun w : ℂ => (w - 1) * riemannZeta w) := by\n    filter_upwards [eventually_ne_nhds hs1] with z hz using zetaA_apply hz\n  have hld : logDeriv zetaA s = 1 / (s - 1) + logDeriv riemannZeta s := by\n    have e1 : logDeriv zetaA s = logDeriv (fun w : ℂ => (w - 1) * riemannZeta w) s := by\n      simp only [logDeriv_apply, hcong.deriv_eq, hcong.self_of_nhds]\n    rw [e1, logDeriv_mul (f := fun w : ℂ => w - 1) (g := riemannZeta) s hs1' hζ\n      ((differentiableAt_id).sub (differentiableAt_const 1))\n      (analyticAt_riemannZeta hs1).differentiableAt]\n    have e2 : logDeriv (fun w : ℂ => w - 1) s = 1 / (s - 1) := by\n      rw [logDeriv_apply]\n      rw [(((hasDerivAt_id s).sub_const (1 : ℂ))).deriv]\n      simp\n    rw [e2]\n  rw [hld]\n  ring\n```\nHmm: `hasDerivAt_id s` : HasDerivAt id 1 s — my function is fun w => w − 1 = id − const: (hasDerivAt_id s).sub_const 1 : HasDerivAt (fun w => w − 1) 1 s? HasDerivAt.sub_const (c) : HasDerivAt (fun x => f x − c) f' x from HasDerivAt f f' x ✓ with f = id: fun w => id w − 1 = fun w => w − 1 ✓ defeq. .deriv : deriv (fun w => w − 1) s = 1 ✓. Then logDeriv_apply: logDeriv f s = deriv f s / f s = 1/(s−1): after rw [logDeriv_apply, that.deriv]: goal (1:ℂ)/(s−1) = 1/(s−1) ✓ rfl/simp... the goal at e2 is logDeriv (fun w => w−1) s = 1/(s−1): rw [logDeriv_apply]: deriv (fun w => w−1) s / (s−1) = 1/(s−1): rw [hderiv]: 1/(s−1) = 1/(s−1) — closes by rfl automatically after rw ✓ (rw closes rfl goals). So drop the simp — or keep `simp` harmless? After rw the goal closes; a trailing simp errors \"no goals\". Write:\n```lean\n    have e2 : logDeriv (fun w : ℂ => w - 1) s = 1 / (s - 1) := by\n      rw [logDeriv_apply, ((hasDerivAt_id s).sub_const (1 : ℂ)).deriv]\n```\nIf the closing rfl isn't syntactic (1/(s−1) = 1/(s−1) IS syntactic ✓) — fine.\n\nThen final: goal logDeriv ζ s = logDeriv zetaA s − 1/(s−1): rw [hld]: logDeriv ζ s = (1/(s−1) + logDeriv ζ s) − 1/(s−1): ring ✓.\n\nAlso Fexpr_eventually_one:\n```lean\nlemma Fexpr_eventually_one :\n    ∃ W : ℂ → ℂ, AnalyticAt ℂ W 1 ∧ ∀ᶠ s in 𝓝[≠] (1 : ℂ), Fexpr s = W s := by\n  have h1dig : ∀ m : ℕ, (1 : ℂ) / 2 ≠ -(m : ℂ) := by\n    intro m h\n    have hre := congrArg Complex.re h\n    simp at hre\n    have : (0:ℝ) ≤ (m : ℝ) := Nat.cast_nonneg m\n    linarith\n  refine ⟨fun s => logDeriv zetaA s - seriesFun s + 1 / s - (Real.log π : ℂ) / 2\n      + Complex.digamma (s / 2) / 2, ?_, ?_⟩\n  · -- analyticity at 1\n    have hA : AnalyticAt ℂ (logDeriv zetaA) 1 :=\n      (zetaA_analyticAt_one.deriv).div zetaA_analyticAt_one (by rw [zetaA_one]; norm_num)\n    have hB : AnalyticAt ℂ seriesFun 1 := seriesFun_analyticAt (by\n      intro h\n      have hlt := h.2.2\n      rw [Complex.one_re] at hlt\n      exact lt_irrefl 1 hlt)\n    have hC : AnalyticAt ℂ (fun s : ℂ => 1 / s) 1 :=\n      analyticAt_const.div analyticAt_id (by norm_num)\n    have hD : AnalyticAt ℂ (fun s : ℂ => (Real.log π : ℂ) / 2) 1 := analyticAt_const\n    have hE : AnalyticAt ℂ (fun s => Complex.digamma (s / 2) / 2) 1 :=\n      analyticAt_digamma_div_two h1dig\n    exact ((((hA.sub hB).add hC).sub hD).add hE)\n  · -- punctured agreement\n    filter_upwards [logDeriv_zeta_eq_logDeriv_zetaA_sub] with s hs\n    rw [Fexpr, hs]\n    ring\n```\nWait — the W function written as `fun s => logDeriv zetaA s - seriesFun s + 1/s - c + digamma(s/2)/2` and the analyticity proof must match that lambda: `((((hA.sub hB).add hC).sub hD).add hE)` : AnalyticAt ℂ (fun s => (((logDeriv zetaA s − seriesFun s) + 1/s) − c) + digamma(s/2)/2)?? AnalyticAt.sub hA hB : AnalyticAt ℂ (fun x => f x − g x)?? AnalyticAt.sub signature: (hf : AnalyticAt f x) (hg : AnalyticAt g x) : AnalyticAt (fun x => f x − g x) x — hmm is it (f − g) as Pi or the lambda? `AnalyticAt.sub : AnalyticAt 𝕜 (f - g) x` — Pi.sub form! Banked gotcha (session 011): \"Pi.mul_apply/Pi.add_apply/Pi.sub_apply simp after .mul/.add/.sub\". So after `exact` the Pi-forms unify with the lambda by defeq (Pi.sub_apply: (f − g) x = f x − g x ✓ defeq) — exact should accept since the goal lambda fun s => ... matches (f − g + ...) applied... The goal: AnalyticAt ℂ (fun s => logDeriv zetaA s - seriesFun s + ...) 1; the term's type: AnalyticAt ℂ (((logDeriv zetaA − seriesFun) + (fun s => 1/s)) − ...) — hmm the exact defeq check: (fun s => f s − g s) vs (f − g): for exact, isDefEq with Pi.sub_apply unfolding — yes works (session-011 gotcha says simp needed AFTER .sub when the GOAL needs it — but exact-unification handles it... the gotcha was about simp rw inside proofs; for exact defeq, (f − g) x ≡ f x − g x by Pi.sub_apply which is defeq ✓). To be safe, add `show` or convert. I'll try direct; if it fails, `convert ... using 2` or funext-rewrite. Actually safest: state W with Pi operations? No — cleaner to keep the lambda and use `refine ... ?_` then `simp only [Pi.sub_apply, Pi.add_apply] at *`... Let me just do:\n```lean\n  refine ⟨fun s => ..., ?_, ?_⟩\n  · exact ((((hA.sub hB).add hC).sub hD).add hE)\n```\nand if defeq fails, adjust. Let me also double-check `analyticAt_const.div analyticAt_id`: AnalyticAt.div (hf : AnalyticAt f x) (hg : AnalyticAt g x) (hg0 : g x ≠ 0): AnalyticAt (f/g) x: for 1/s: f = const 1, g = id: analyticAt_const.div analyticAt_id (show (id 1 : ℂ) ≠ 0...) — g x = id 1 = 1: `(by norm_num : (1:ℂ) ≠ 0)` — the hg0 type: id 1 ≠ 0 — norm_num proves id 1 ≠ 0? id 1 reduces to 1 ✓ norm_num handles. Hmm but the expected type might be `(fun x => x) 1 ≠ 0` — also fine.\n\nFor the ring at the end: goal: logDeriv ζ s − seriesFun s + 1/s + 1/(s−1) − c + ψ/2 = logDeriv zetaA s − seriesFun s + 1/s − c + ψ/2 with hs : logDeriv ζ s = logDeriv zetaA s − 1/(s−1): after rw [Fexpr, hs] — wait rw [Fexpr] unfolds? Fexpr is a def — `rw [Fexpr]`?? same issue as before — use `simp only [Fexpr]` or `show`. The goal is Fexpr s = W s where W is the lambda: (fun s => ...) s — beta-redex! After `refine ⟨fun s => ..., ?_, ?_⟩` the second goal is ∀ᶠ s in 𝓝[≠] 1, Fexpr s = (fun s' => ...) s — the RHS is a redex. filter_upwards [...] with s hs: goal Fexpr s = (fun ...) s. `rw [hs]`? pattern logDeriv riemannZeta s inside Fexpr — Fexpr is a def: rw won't see through. Use: `simp only [Fexpr]` first (unfolds via equation lemma), then rw [hs], then the redex: `show`-beta: simp only [] reduces? Add `show logDeriv riemannZeta s - seriesFun s + 1/s + 1/(s-1) - (Real.log π:ℂ)/2 + Complex.digamma (s/2)/2 = logDeriv zetaA s - seriesFun s + 1/s - (Real.log π:ℂ)/2 + Complex.digamma (s/2)/2`? The RHS redex beta-reduces to exactly that ✓ show accepts (defeq). Then rw [hs]; ring.\n\nHmm — rw [hs] where hs : logDeriv riemannZeta s = logDeriv zetaA s − 1/(s−1): rewrites LHS occurrence ✓ then ring: (A − 1/(s−1)) − S + 1/s + 1/(s−1) − c + d = A − S + 1/s − c + d ✓ ring (divisions as atoms: 1/(s−1) appears twice identically ✓ cancels).\n\nNow — also there's a name collision risk: `badPoint`, `Fexpr`, `analyticAt_digamma` — check no existing decls with these names in the package. analyticAt_digamma — LiDigamma has digamma_analyticOn (different name) ✓. badPoint — new ✓. Fexpr — LiSkeleton has FAlt, Ffactor... Fexpr new ✓ (there's `Ffun` in LiSides? and `FfunT`? — Fexpr distinct ✓).\n\nAlso `zeta_zero_classification` — check RvM names: `RvM.completedRiemannZeta_eq_Gammaℝ_mul` and `RvM.completedRiemannZeta_eq_zero_iff` — the RvM namespace: LiDisc used `RvM.completedRiemannZeta_eq_Gammaℝ_mul hρ0 hG` ✓ and `RvM.completedRiemannZeta_eq_zero_iff.mp` ✓ — with `open Zeta23` so RvM accessible ✓.\n\n`Complex.Gammaℝ_eq_zero_iff` ✓ (LiDisc used). `riemannZeta_ne_zero_of_one_le_re` ✓.\n\nNow the m=0 case in classification: hn : s = -(2 * ((0:ℕ) : ℂ)): `simp at hn` — simp normalizes: Nat.cast_zero, mul_zero... wait: -(2 * ↑(0:ℕ)): ↑0 = 0, 2*0 = 0, -0 = 0: hn : s = 0 ✓ then absurd hn hs0 ✓. But careful — `simp at hn` might rewrite hn : s = 0 and then... it stays as hypothesis hn : s = 0 (simp doesn't close since goal isn't affected)... then `exact absurd hn hs0` ✓.\n\nHmm, one thing in the classification: Gammaℝ_eq_zero_iff gives `∃ m : ℕ, s = -(2 * ↑m)` — the exact form from the gotcha \"−(2*↑m)\". obtain ⟨n, hn⟩ : hn : s = -(2 * (n:ℂ))?? Then my third disjunct wants s = -2 * (n:ℂ): `by rw [hn]; ring` ✓.\n\nAlso `Nat.eq_zero_or_pos` ✓ used before.\n\nLet me now rewrite §4 properly and compile §0–§4.","signature":""},{"type":"text","text":"Let me fix §4 properly before the first compile."},{"type":"tool_use","id":"Edit_114","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean","old_string":"/-- Near s = 1 (punctured): logDeriv ζ s = logDeriv zetaA s − 1/(s−1). -/\nlemma logDeriv_zeta_eq_logDeriv_zetaA_sub :\n    ∀ᶠ s in 𝓝[≠] (1 : ℂ),\n      logDeriv riemannZeta s = logDeriv zetaA s - 1 / (s - 1) := by\n  filter_upwards [self_mem_nhdsWithin,\n    mem_nhdsWithin_of_mem_nhds zetaA_ne_zero_near_one,\n    mem_nhdsWithin_of_mem_nhds (eventually_ne_nhds (show (1 : ℂ) ≠ 1 from by norm_num))]\n    with s hs1 hZA hne1\n  · -- this branch is unreachable: the third filter is trivial; see below\n  · skip\n\n/-- The witness at s = 1: Fexpr agrees punctured with\nlogDeriv zetaA − seriesFun + 1/s − (log π)/2 + ψ(s/2)/2. -/\nlemma Fexpr_eventually_one :\n    ∃ W : ℂ → ℂ, AnalyticAt ℂ W 1 ∧ ∀ᶠ s in 𝓝[≠] (1 : ℂ), Fexpr s = W s := by\n  sorry","new_string":"/-- Near s = 1 (punctured): logDeriv ζ s = logDeriv zetaA s − 1/(s−1). -/\nlemma logDeriv_zeta_eq_logDeriv_zetaA_sub :\n    ∀ᶠ s in 𝓝[≠] (1 : ℂ),\n      logDeriv riemannZeta s = logDeriv zetaA s - 1 / (s - 1) := by\n  filter_upwards [self_mem_nhdsWithin,\n    mem_nhdsWithin_of_mem_nhds zetaA_ne_zero_near_one] with s hs1 hZA\n  have hs1' : s - 1 ≠ 0 := sub_ne_zero.mpr hs1\n  have hζ : riemannZeta s ≠ 0 := by\n    have e := zetaA_apply hs1\n    intro hz\n    rw [hz, mul_zero] at e\n    exact hZA e.symm\n  have hcong : zetaA =ᶠ[𝓝 s] (fun w : ℂ => (w - 1) * riemannZeta w) := by\n    filter_upwards [eventually_ne_nhds hs1] with z hz using zetaA_apply hz\n  have hld : logDeriv zetaA s = 1 / (s - 1) + logDeriv riemannZeta s := by\n    have e1 : logDeriv zetaA s = logDeriv (fun w : ℂ => (w - 1) * riemannZeta w) s := by\n      simp only [logDeriv_apply, hcong.deriv_eq, hcong.self_of_inds]\n    rw [e1, logDeriv_mul (f := fun w : ℂ => w - 1) (g := riemannZeta) s hs1' hζ\n      (differentiableAt_id.sub (differentiableAt_const 1))\n      (analyticAt_riemannZeta hs1).differentiableAt]\n    have e2 : logDeriv (fun w : ℂ => w - 1) s = 1 / (s - 1) := by\n      rw [logDeriv_apply, ((hasDerivAt_id s).sub_const (1 : ℂ)).deriv]\n    rw [e2]\n  rw [hld]\n  ring\n\n/-- The witness at s = 1: Fexpr agrees punctured with\nlogDeriv zetaA − seriesFun + 1/s − (log π)/2 + ψ(s/2)/2, analytic at 1. -/\nlemma Fexpr_eventually_one :\n    ∃ W : ℂ → ℂ, AnalyticAt ℂ W 1 ∧ ∀ᶠ s in 𝓝[≠] (1 : ℂ), Fexpr s = W s := by\n  have h1dig : ∀ m : ℕ, (1 : ℂ) / 2 ≠ -(m : ℂ) := by\n    intro m h\n    have hre := congrArg Complex.re h\n    simp at hre\n    have hm : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg m\n    linarith\n  refine ⟨fun s => logDeriv zetaA s - seriesFun s + 1 / s - (Real.log π : ℂ) / 2\n      + Complex.digamma (s / 2) / 2, ?_, ?_⟩\n  · have hA : AnalyticAt ℂ (logDeriv zetaA) 1 :=\n      (zetaA_analyticAt_one.deriv).div zetaA_analyticAt_one (by rw [zetaA_one]; norm_num)\n    have hB : AnalyticAt ℂ seriesFun 1 := seriesFun_analyticAt (by\n      intro h\n      have hlt := h.2.2\n      rw [Complex.one_re] at hlt\n      exact lt_irrefl 1 hlt)\n    have hC : AnalyticAt ℂ (fun s : ℂ => 1 / s) 1 :=\n      analyticAt_const.div analyticAt_id (by norm_num)\n    have hD : AnalyticAt ℂ (fun s : ℂ => (Real.log π : ℂ) / 2) 1 := analyticAt_const\n    have hE : AnalyticAt ℂ (fun s => Complex.digamma (s / 2) / 2) 1 :=\n      analyticAt_digamma_div_two h1dig\n    exact ((((hA.sub hB).add hC).sub hD).add hE)\n  · filter_upwards [logDeriv_zeta_eq_logDeriv_zetaA_sub] with s hs\n    show logDeriv riemannZeta s - seriesFun s + 1 / s + 1 / (s - 1)\n        - (Real.log π : ℂ) / 2 + Complex.digamma (s / 2) / 2\n      = logDeriv zetaA s - seriesFun s + 1 / s - (Real.log π : ℂ) / 2\n        + Complex.digamma (s / 2) / 2\n    rw [hs]\n    ring"},"extra_content":null}]