[{"type":"thinking","thinking":"1. `analyticOrderAt_ne_top_of_isPreconnected` is inside `namespace AnalyticOnNhd` (line 569-706) — so full name: `AnalyticOnNhd.analyticOrderAt_ne_top_of_isPreconnected`. Also from the earlier grep at line 627: `theorem analyticOrderAt_ne_top_of_isPreconnected {x y : 𝕜} (hf : AnalyticOnNhd 𝕜 f U) (hU : IsPreconnected U) (h₁x : x ∈ U) (hy : y ∈ U) (h₂x : analyticOrderAt f x ≠ ⊤) : analyticOrderAt f y ≠ ⊤` — wait let me double check the argument order: (hf) (hU) (h₁x : x ∈ U) (hy : y ∈ U) (h₂x). From the earlier read: `(hf : AnalyticOnNhd 𝕜 f U) (hU : IsPreconnected U) (h₁x : x ∈ U) (hy : y ∈ U) (h₂x : analyticOrderAt f x ≠ ⊤)`. Hmm the earlier output showed:\n\n```\ntheorem analyticOrderAt_ne_top_of_isPreconnected {x y : 𝕜} (hf : AnalyticOnNhd 𝕜 f U)\n    (hU : IsPreconnected U) (h₁x : x ∈ U) (hy : y ∈ U) (h₂x : analyticOrderAt f x ≠ ⊤) :\n    analyticOrderAt f y ≠ ⊤ :=\n```\n✓ So: AnalyticOnNhd.analyticOrderAt_ne_top_of_isPreconnected zetaA_analyticOnNhd_univ isPreconnected_univ (Set.mem_univ 1) (Set.mem_univ ρ) (proof order zetaA 1 ≠ ⊤). With x := 1, y := ρ ✓.\n\nAlso `exists_analyticOrderAt_ne_top_iff_forall` is also in AnalyticOnNhd namespace presumably ✓ not needed.\n\n2. `mul_div_cancel_left₀ (b : M₀) {a : M₀} (ha : a ≠ 0) : a * b / a = b` — the explicit arg is b (the \"other\" factor), a implicit. So: `mul_div_cancel_left₀ _ (pow_ne_zero (n + 1) hs1n)` — with b := (s−1)^k * h s inferred... wait b explicit: mul_div_cancel_left₀ ((s−1)^k * h s)?? — if I pass `_` Lean infers b from the match in rw — rw elaborates the rule with metavariables then matches the pattern a * b / a — passing _ for b and the proof for a ✓: `mul_div_cancel_left₀ _ (pow_ne_zero (n + 1) hs1n)` : ?a * ?b / ?a = ?b with ?a := (s−1)^{n+1} from the proof ✓ then rw matches ✓.\n\nNow fix all:\n\n**Fix 1** (zeta_order_ne_top_of_ne): full rewrite of the lemma:\n```lean\nlemma zeta_order_ne_top_of_ne {ρ : ℂ} (hρ : ρ ≠ 1) :\n    analyticOrderAt riemannZeta ρ ≠ ⊤ := by\n  have h0 : analyticOrderAt zetaA 1 = 0 :=\n    (zetaA_analyticAt_one.analyticOrderAt_eq_zero).mpr (by rw [zetaA_one]; norm_num)\n  have hA : analyticOrderAt zetaA ρ ≠ ⊤ :=\n    AnalyticOnNhd.analyticOrderAt_ne_top_of_isPreconnected zetaA_analyticOnNhd_univ\n      isPreconnected_univ (Set.mem_univ 1) (Set.mem_univ ρ) (by rw [h0]; simp)\n  have hcong : zetaA =ᶠ[𝓝 ρ] (fun w : ℂ => w - 1) * riemannZeta := by\n    filter_upwards [eventually_ne_nhds hρ] with w hw using zetaA_apply hw\n  rw [analyticOrderAt_congr hcong,\n    analyticOrderAt_mul (f := fun w : ℂ => w - 1) (g := riemannZeta)\n      (analyticAt_id.sub analyticAt_const) (analyticAt_riemannZeta hρ)] at hA\n  have h0' : analyticOrderAt (fun w : ℂ => w - 1) ρ = 0 :=\n    ((show AnalyticAt ℂ (fun w : ℂ => w - 1) ρ from\n      analyticAt_id.sub analyticAt_const).analyticOrderAt_eq_zero).mpr (sub_ne_zero.mpr hρ)\n  rw [h0', zero_add] at hA\n  exact hA\n```\nConcerns:\n- hcong's pointwise: goal zetaA w = ((fun w => w−1) * riemannZeta) w — defeq (w−1)*ζ w — using zetaA_apply hw : zetaA w = (w−1)*ζ w ✓ defeq through Pi.mul_apply — filter_upwards `using` checks defeq ✓ hopefully. If not: add (show zetaA w = (w−1)*riemannZeta w from zetaA_apply hw)... the `using` term's type must defeq-match the goal — Pi.mul_apply reduction is defeq ✓.\n- rw [analyticOrderAt_congr hcong] at hA: hA : analyticOrderAt zetaA ρ ≠ ⊤ — pattern analyticOrderAt zetaA ρ — the lemma: analyticOrderAt_congr (hfg : f =ᶠ[𝓝 z₀] g) : analyticOrderAt f z₀ = analyticOrderAt g z₀ — rewriting at hA: replaces analyticOrderAt zetaA ρ with analyticOrderAt ((fun w => w−1) * riemannZeta) ρ ✓.\n- rw [analyticOrderAt_mul ...] at hA: the explicit (f :=) (g :=) force the pattern: analyticOrderAt_mul's LHS pattern analyticOrderAt (f * g) ρ with f := fun w => w−1, g := riemannZeta — matches hA's term ✓ — becomes analyticOrderAt (fun w => w−1) ρ + analyticOrderAt riemannZeta ρ ≠ ⊤ ✓.\n  The proof args: analyticAt_id.sub analyticAt_const : AnalyticAt ℂ (id − fun x => ?) — expected AnalyticAt ℂ (fun w => w−1) ρ — defeq ✓.\n- rw [h0', zero_add] at hA ✓ → analyticOrderAt riemannZeta ρ ≠ ⊤ ✓.\n\nWait — analyticOrderAt_mul's f/g are implicit? `theorem analyticOrderAt_mul (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) : analyticOrderAt (f * g) z₀ = analyticOrderAt f z₀ + analyticOrderAt g z₀` — f g implicit (inferred from hf hg). With (f := ...) (g := ...) named — fine if they're implicit named args ✓.\n\n**Fix 2** (ppart_mul_pow key): remove trailing `ring` (field_simp closed it). Actually — WAIT. Let me recheck: line 216 error \"No goals to be solved\" — the `ring` after field_simp in key. But is it certain field_simp closed it? The error says no goals at the ring — yes field_simp closed it. Remove ring. But hold on — is it possible field_simp REORDERED leaving something ring closes... no, \"no goals\" at line 216 = ring's location. Remove `ring` in key. \n\n**Fix 3** (Jlag_mul_pow): remove trailing `ring` (line 232). Hmm — but wait, is that safe? If field_simp closed the goal, removing ring is right. The error at 232:2 \"No goals to be solved\" — Jlag_mul_pow's ring ✓ remove.\n\nAlso Jlag_mul_pow and ppart_mul_pow: make n explicit. Update signature lines and the call in Rfun_eq_sub_div_near_one: `Jlag_mul_pow n hs1' hs0`, `ppart_mul_pow n hs1'`.\n\n**Fix 4** (Rfun_tendsto_one): \n- ⊤-case: ascribe hev as EventuallyEq: \n```lean\n    have hev : (Rfun n) =ᶠ[𝓝[≠] (1 : ℂ)] (fun _ => (0 : ℂ)) := by\n      filter_upwards [Rfun_eq_sub_div_near_one n, mem_nhdsWithin_of_mem_nhds h0] with s hs hg0\n      have hg0' : Ffun n s - TaylP n s = 0 := hg0\n      rw [hs, hg0']\n      exact zero_div _\n    exact tendsto_const_nhds.congr' hev.symm\n```\nWait — with the EventuallyEq ascription, the filter_upwards goal: (fun s => Rfun n s = (fun _ => 0) s) — i.e. Rfun n s = 0 after beta ✓ same proof ✓. tendsto_const_nhds.congr' hev.symm — hmm direction again: Tendsto.congr' (hl : f₁ =ᶠ[l] f₂) (h : Tendsto f₁ l l₂) : Tendsto f₂ l l₂. I have hev : Rfun n =ᶠ (fun _ => 0); hev.symm : (fun _ => 0) =ᶠ Rfun n; tendsto_const_nhds : Tendsto (fun _ => 0) _ (𝓝 0). So hl := hev.symm (f₁ := fun _ => 0, f₂ := Rfun n), h := tendsto_const_nhds ✓ gives Tendsto (Rfun n) (𝓝[≠] 1) (𝓝 0) ✓. `tendsto_const_nhds.congr' hev.symm` — dot notation: tendsto_const_nhds is the Tendsto arg (second explicit) — dot inserts at first matching explicit type — hl is EventuallyEq (not Tendsto) so inserts at h ✓.\n\nHmm — actually wait: is it `.congr'` with the EventuallyEq FIRST? Tendsto.congr' signature: `theorem Filter.Tendsto.congr' {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (hl : f₁ =ᶠ[l₁] f₂) (h : Tendsto f₁ l₁ l₂) : Tendsto f₂ l₁ l₂` ✓ hl first ✓.\n\n- m-case: ascribe hev as EventuallyEq:\n```lean\n    have hev : (Rfun n) =ᶠ[𝓝[≠] (1 : ℂ)]\n        (fun s => (s - 1) ^ (analyticOrderNatAt (Ffun n - TaylP n) 1 - (n + 1)) * h s) := by\n      filter_upwards ...\n      -- goal: Rfun n s = (s−1)^k * h s\n```\nThen final: `(hcont.tendsto.mono_left nhdsWithin_le_nhds).congr' hev.symm` — hev.symm : φ =ᶠ Rfun n; hcont.tendsto... : Tendsto φ (𝓝[≠] 1) (𝓝 (φ 1)) — φ 1 = (1−1)^k·h 1 — the refine witness ⟨(1−1)^k * h 1, ...⟩ ✓.\n\nHmm wait — actually the witness: Tendsto φ (𝓝[≠] 1) (𝓝 (φ 1)) where φ 1 beta-reduces to (1−1)^k * h 1 — the refine's witness is that expression ✓ defeq ✓.\n\n- Fix the e/mul_assoc/mul_div_cancel_left₀ rewrite: `rw [e, mul_assoc, mul_div_cancel_left₀ _ (pow_ne_zero (n + 1) hs1n)]` — hmm mul_assoc rewrites a·b·c = a·(b·c) — first occurrence: the numerator ((s−1)^{n+1} * (s−1)^k) * h s → (s−1)^{n+1} * ((s−1)^k * h s) ✓ then mul_div_cancel_left₀ _ (proof) : ?a * ?b / ?a = ?b with ?a := (s−1)^{n+1} — matches `(s−1)^{n+1} * ((s−1)^k * h s) / (s−1)^{n+1}` ✓ → (s−1)^k * h s — then goal RHS is (s−1)^k * h s — rfl closed by rw ✓.\n\n**Fix 5** (Rfun_residue_at hexp's hfhs): coerce to EventuallyEq:\n```lean\n    have hfhs' : riemannZeta =ᶠ[𝓝 s]\n        (fun z => (z - ρ) ^ analyticOrderNatAt riemannZeta ρ • h z) := hfhs\n```\nthen use hfhs' in e1's simp only. Hmm — will the ascription work? hfhs : {x | riemannZeta x = (x−ρ)^m • h x} ∈ 𝓝 s (the error display). EventuallyEq riemannZeta (fun z => ...) (𝓝 s) := ∀ᶠ x in 𝓝 s, riemannZeta x = (fun z => ...) x — Filter.Eventually (𝓝 s) (fun x => ...) := (𝓝 s).Eventually (fun x => ...) — is `∀ᶠ x in l, p x` defeq to `{x | p x} ∈ l`? Eventually is DEFINED as `l.Eventually p := {x | p x} ∈ l`? Filter.Eventually (l : Filter α) (p : α → Prop) : Prop := {x | p x} ∈ l — YES defeq ✓ and EventuallyEq f g l := ∀ᶠ x in l, f x = g x — defeq ✓ with beta on the RHS: (fun z => ...) x ⟶ (x−ρ)^m • h x ✓ defeq. So the ascription `:= hfhs` typechecks ✓.\n\nAlso — the error at 386:84 \"unsolved goals n ρ\" — the e1 have — cascade. And line 387's hfhs.deriv_eq error — fixed by hfhs'.\n\nWait — also check: in e1 I wrote `logDeriv (fun z => (z - ρ) ^ analyticOrderNatAt riemannZeta ρ * h z) s` — the mul form (not •) — the simp only [logDeriv_apply, hfhs'.deriv_eq, hfhs'.self_of_nhds, smul_eq_mul]: logDeriv_apply unfolds logDeriv riemannZeta s → deriv ζ s/ζ s; hfhs'.deriv_eq : deriv riemannZeta s = deriv (fun z => (z−ρ)^m • h z) s; hfhs'.self_of_nhds : riemannZeta s = (s−ρ)^m • h s; smul_eq_mul : •→*. After: deriv (fun z => (z−ρ)^m * h z)... hmm — deriv_eq gives deriv (fun z => (z−ρ)^m • h z) — with smul_eq_mul also rewriting the • inside deriv's argument ✓ (simp works under binders ✓). Then the two sides become equal: deriv (•-form)/•-form = deriv (*-form)/*-form — after smul_eq_mul everywhere ✓ syntactically equal → simp closes by rfl? The goal e1: logDeriv ζ s = logDeriv (fun z => (z−ρ)^m * h z) s — after simp only rewrites LHS: deriv (•-form) s / (•-form) s — and smul_eq_mul turns •-forms into *-forms: LHS = deriv (fun z => (z−ρ)^m * h z) s / (fun z => (z−ρ)^m * h z) s; RHS: logDeriv (*-form) s — logDeriv_apply also rewrites RHS → deriv (*-form) s / (*-form) s ✓ equal → rfl ✓ simp closes. Vendored did exactly this ✓.\n\nHmm — one question: in e1, hfhs'.deriv_eq — EventuallyEq.deriv_eq : f =ᶠ[𝓝 x] g → deriv f x = deriv g x ✓ vendored used hfs.deriv_eq ✓.\n\n**Additional fix** — the hld's `rw [e2]` at the end: after rw [e1, logDeriv_mul ...], goal: logDeriv (fun z => (z−ρ)^m) s + logDeriv h s = m/(s−ρ) + logDeriv h s — rw [e2] rewrites logDeriv (fun z => (z−ρ)^m) s → m/(s−ρ) — both sides equal — auto rfl ✓.\n\nBut WAIT — e2's trailing `simp`: after rw [show ... from rfl, logDeriv_comp (by fun_prop) (by fun_prop), logDeriv_pow], what's the exact goal? logDeriv_comp : logDeriv (g ∘ f) x = logDeriv g (f x) * deriv f x? or (deriv f x) * logDeriv g (f x)? — some commuted form; logDeriv_pow : logDeriv (fun x => x^k) x = (k : 𝕜)/x? Then: ((m:ℂ)/((s−ρ))) * deriv (fun z => z−ρ) s = m/(s−ρ) — deriv (fun z => z−ρ) s = 1 — simp [deriv_sub_const, deriv_id''] ✓ plain `simp` handles (vendored's `simp` ✓ copy).\n\nHmm — logDeriv_comp's exact conclusion form might be `logDeriv (g ∘ f) x = logDeriv g (f x) * deriv f x` — if instead it's deriv f x * logDeriv g (f x) — either way simp normalizes ✓.\n\nNow also double-check `ppart_differentiableAt`'s first rw: `rw [show (ppart n) = ∑ i ∈ Finset.Icc 1 (n+1), (fun w : ℂ => Acoeff n i / (w−1)^i) from by funext w; simp [ppart, Finset.sum_apply]]` — the funext: ppart n w = (∑ i ∈ Icc, (fun w => Acoeff n i/(w−1)^i)) w — RHS beta: ∑ i ∈ Icc, (fun w => ...) w — Finset.sum_apply : (∑ i ∈ s, f i) x = ∑ i ∈ s, f i x ✓ so RHS = ∑ i ∈ Icc, Acoeff n i/(w−1)^i = ppart n w by def — simp [ppart, Finset.sum_apply] closes ppart n w = (Σ-lambdas) w? goal after funext w: ppart n w = (∑ i ∈ Finset.Icc 1 (n+1), fun w => Acoeff n i / (w-1)^i) w — simp [Finset.sum_apply] RHS → ∑ i, Acoeff n i/(w−1)^i; simp [ppart] LHS → ∑ i ∈ Icc 1 (n+1), Acoeff n i/(w−1)^i ✓ equal ✓. Hmm — Gprim_differentiableAt used `unfold Gprim; rw [show ...]` — I used rw [show (ppart n) = ...]: rewriting the FUNCTION ppart n in the goal `DifferentiableAt ℂ (ppart n) s` ✓ fine.\n\nSimilarly TaylP_analyticAt's `exact this _` — this (range (n+1)) : AnalyticAt ℂ (fun w => ∑ j ∈ range(n+1), ...) w — goal AnalyticAt ℂ (TaylP n) w — defeq delta+beta ✓.\n\nAnd in iteratedDeriv_TaylP I wrote `rw [show (TaylP n) = ∑ j ∈ ..., (fun w => ...) from by funext w; simp [TaylP, Finset.sum_apply]]` ✓ analogous.\n\nNow — ppart_continuousAt's `exact this _`: goal ContinuousAt (ppart n) s; this (Icc 1 (n+1)) : ContinuousAt (fun w => ∑ i ∈ Icc 1 (n+1), Acoeff n i/(w−1)^i) s — defeq ✓.\n\nOne more possible issue — in `ppart_differentiableAt` I dropped `unfold` and used the show-rw directly on (ppart n) — fine.\n\nAlso `ppart_mul_pow`'s `show`: goal `(s - 1) ^ (n + 1) * ppart n s = TaylP n s` — show with unfolded ppart/TaylP ✓ defeq.\n\nAnd Jlag_mul_pow's `show (s - 1) ^ (n + 1) * (kFun n s * ...) = (s ^ n - (s - 1) ^ n) * hLi s` ✓ defeq (Jlag/Ffun unfolds).\n\nAlso `Rfun_eq_sub_div_near_one`'s e: `show (s - 1) ^ (n + 1) * (Jlag n s - ppart n s) = Ffun n s - TaylP n s` — the stated e has `(s-1)^{n+1} * Rfun n s` — Rfun n s defeq Jlag n s − ppart n s (delta Rfun + Pi.sub_apply) — show ✓.\n\nThen `rw [mul_sub, e1, e2]` — mul_sub : a*(b−c) = a*b − a*c ✓ rewrites LHS → (s−1)^{n+1} * Jlag n s − (s−1)^{n+1} * ppart n s — then e1 : (s−1)^{n+1} * Jlag n s = Ffun n s ✓ rw, e2 ✓ rw → Ffun n s − TaylP n s = Ffun n s − TaylP n s — rfl auto ✓.\n\nNow — the LAST potential issue in rectInt'_Jlag: `simpa [hAdef] using Rfun_bounded_near_one n` — the goal: ((Rfun n) − fun s => A 1 / (s - 1)) =O[𝓝[≠] 1] (1 : ℂ → ℂ). simp [hAdef]: A → fun p => if p = 1 then 0 else ...; A 1 → if 1 = 1 then 0 else ... → simp if_pos → 0 — wait, the `set` made A a local def with hAdef : A = fun p => ... — simp [hAdef] rewrites A occurrences ✓. Then (fun s => (0:ℂ)/(s−1)): zero_div simp ✓ → fun s => 0 → = 0? (fun s => (0:ℂ)) = (0 : ℂ → ℂ)? Pi.zero — simp [Pi.zero_apply]? — the lambda fun s => 0 IS Pi.zero applied... defeq — simp normalizes both. Then Rfun n − 0 → sub_zero ✓ Rfun n. So goal becomes (Rfun n) =O[𝓝[≠] 1] 1 ✓ = Rfun_bounded_near_one n ✓ simpa closes. Risk: moderate; fallback: \n```lean\n      · have hA1 : A 1 = 0 := by simp [hAdef]\n        have h2 : ((Rfun n) - fun s => A 1 / (s - 1)) = Rfun n := by\n          rw [hA1]; simp\n        rw [h2]\n        exact Rfun_bounded_near_one n\n```\nI'll preemptively use this explicit version (safer).\n\nSimilarly the hAp simp [hAdef, hp1] — if p = 1 condition with hp1 : p ≠ 1 — simp resolves if_neg ✓ fine.\n\nhsum's `simp [hAdef, hρ1]` — goal A ρ = (zeroMult ρ:ℂ)*(1−(ρ/(ρ−1))^n) — simp [hAdef] unfolds A ρ → if ρ = 1 then 0 else formula → if_neg hρ1 ✓ → formula ✓ closes by rfl ✓.\n\nAlso in fHolo — `hs.2` : s ∉ ↑(insert 1 Sz) — wait: hs : s ∈ Rectangle z w \\ ↑(insert (1:ℂ) Sz) — hs.2 : s ∉ (↑(insert 1 Sz) : Set ℂ) ✓ my hsc ✓. Then `Finset.mem_coe.mpr (h ▸ Finset.mem_insert_self 1 Sz)` — Finset.mem_insert_self 1 Sz : 1 ∈ insert 1 Sz — h ▸ ... : h : s = 1 — rewriting 1 → s?? h ▸ x rewrites h's LHS→RHS? h : s = 1; h ▸ (proof about 1)... ▸ rewrites occurrences of s (LHS) by 1 (RHS) or vice versa depending on elaboration — ambiguous. Safer: `h.symm ▸ Finset.mem_insert_self 1 Sz` : h.symm : 1 = s — ▸ rewrites 1 to s in Finset.mem_insert_self 1 Sz's type: 1 ∈ insert 1 Sz → s ∈ insert 1 Sz ✓ then Finset.mem_coe.mpr : s ∈ ↑(insert 1 Sz) ✓. Let me use that. Or simply: `Finset.mem_coe.mpr (Finset.mem_insert_self 1 Sz)` has type 1 ∈ ↑(...) — need s ∈ — with h : s = 1: `h ▸ ...` — Lean's ▸ : h : a = b → motive a → motive b (rewrites a→b) — h : s = 1, term : 1 ∈ insert... — h ▸ term : tries to rewrite s → 1 in term's type (1 ∈ ... has no s) or 1 → s — the ▸ elaboration with the expected type s ∈ ↑(...) — given the expected type, ▸ picks the right direction — with `fun h => hsc (h ▸ ...)` the expected type guides ✓ usually works. To be safe write: `(Finset.mem_coe.mpr (h.symm ▸ Finset.mem_insert_self 1 Sz))` ✓ deterministic.\n\nHmm wait — h ▸ e where h : s = 1, e : 1 ∈ insert 1 Sz — ▸ is heterogeneous: h ▸ e rewrites the TYPE of e along h: it abstracts occurrences of the LHS of h in type... ugh, direction confusion. h.symm ▸ e with h.symm : 1 = s — rewrites 1 ↦ s in e's type? ▸ in Lean 4: `h ▸ e` where h : a = b — replaces a with b in the type of e... no wait — `Eq.mpr` style: h ▸ e : the motive is inferred from the expected type; occurrences of a are replaced by b. h.symm : 1 = s — replaces 1 by s ✓ gives s ∈ insert 1 Sz ✓ deterministic given expected type. OK use h.symm ▸.\n\nAlso in hSz_ne_one : fun ρ hρ h => h1notSz (h ▸ hρ) — h : ρ = 1, hρ : ρ ∈ Sz — want 1 ∈ Sz: h ▸ hρ rewrites ρ→1 ✓ (h : ρ = 1, ▸ replaces ρ by 1 ✓) — deterministic enough ✓ keep.\n\nIn hRint: `not_mem_rectangleBorder_of_rectangle_mem_nhds h1 (h ▸ hs)` — h : s = 1, hs : s ∈ RectangleBorder z w — want 1 ∈ border: h ▸ hs ✓ same pattern.\n\nOK — and `hSz_zeta`'s `rcases h2 with h | h; · exact absurd (sub_eq_zero.mp h) hρ1; · exact h` — h2 after rw [zetaA_apply hρ1, mul_eq_zero]: (ρ−1) * ζ ρ = 0 becomes ρ − 1 = 0 ∨ ζ ρ = 0 — mul_eq_zero : a*b = 0 ↔ a = 0 ∨ b = 0 — rw at h2 turns h2 into the ∨ ✓ (mul_eq_zero is an iff — rw at hypothesis with iff rewrites ✓).\n\nWait — rw [zetaA_apply hρ1] at h2: h2 : zetaA ρ = 0 → becomes (ρ−1)*ζ ρ = 0 ✓ then rw [mul_eq_zero] at h2: h2 : ρ − 1 = 0 ∨ riemannZeta ρ = 0 ✓.\n\nabsurd (sub_eq_zero.mp h) hρ1: sub_eq_zero.mp h : ρ = 1; hρ1 : ρ ≠ 1 — absurd (ρ = 1) (ρ ≠ 1) : C ✓ (goal is riemannZeta ρ = 0 — any C ✓).\n\nNow let me also double check `Jlag_differentiableAt`'s `.neg` placement: (hζ.deriv.differentiableAt.div hζ.differentiableAt hs0) : DifferentiableAt ℂ (fun w => deriv ζ w/ζ w) s? — .div gives DifferentiableAt ((deriv ζ) / ζ) — Pi.div — the target lambda fun w => deriv ζ w/ζ w — defeq ✓. .neg : DifferentiableAt (-(deriv ζ/ζ)) — Pi.neg — vs fun w => −(deriv ζ w/ζ w) defeq ✓. Then (kFun_differentiableAt hs1).mul hd : DifferentiableAt (kFun n * (fun w => −(deriv ζ w/ζ w))) — Pi.mul — vs Jlag n = fun s => kFun n s * (−(deriv ζ s/ζ s)) — defeq ✓.\n\nAlso Jlag_differentiableAt — I wrote:\n```lean\n  have hd : DifferentiableAt ℂ (fun w : ℂ => -(deriv riemannZeta w / riemannZeta w)) s :=\n    (hζ.deriv.differentiableAt.div hζ.differentiableAt hs0).neg\n```\nhζ.deriv : AnalyticAt ℂ (deriv riemannZeta) s ✓ .differentiableAt ✓.\n\nOK let me now apply all fixes. Files edits:\n\n1. zeta_order_ne_top_of_ne — replace whole proof.\n2. ppart_mul_pow — make n explicit (signature `(n : ℕ) {s : ℂ}`); remove trailing ring in key.\n3. Jlag_mul_pow — make n explicit; remove trailing ring.\n4. Rfun_eq_sub_div_near_one — update calls `Jlag_mul_pow n hs1' hs0`, `ppart_mul_pow n hs1'`.\n5. Rfun_tendsto_one — EventuallyEq ascriptions ×2; fix mul_div_cancel_left₀ arg.\n6. Rfun_residue_at — add hfhs' coercion; use in e1.\n7. rectInt'_Jlag — explicit hA1/h2 instead of simpa (keep simpa as-is? I decided explicit version — let me use explicit).\n8. fHolo's h ▸ → h.symm ▸.\n\nLet me make these edits.","signature":""},{"type":"text","text":"All errors are mechanical. Applying fixes."},{"type":"tool_use","id":"Edit_35","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","old_string":"  have hA : analyticOrderAt zetaA ρ ≠ ⊤ :=\n    analyticOrderAt_ne_top_of_isPreconnected zetaA_analyticOnNhd_univ\n      isPreconnected_univ (Set.mem_univ 1) (Set.mem_univ ρ) (by rw [h0]; simp)\n  have hcong : zetaA =ᶠ[𝓝 ρ] (fun w : ℂ => (w - 1) * riemannZeta w) := by\n    filter_upwards [eventually_ne_nhds hρ] with w hw using (zetaA_apply hw).symm\n  rw [analyticOrderAt_congr hcong,\n    show (fun w : ℂ => (w - 1) * riemannZeta w) =\n      (fun w : ℂ => w - 1) * riemannZeta from by funext w; simp [Pi.mul_apply],\n    analyticOrderAt_mul (analyticAt_id.sub analyticAt_const) (analyticAt_riemannZeta hρ),\n    ((analyticAt_id.sub analyticAt_const).analyticOrderAt_eq_zero).mpr (sub_ne_zero.mpr hρ),\n    zero_add] at hA\n  exact hA","new_string":"  have hA : analyticOrderAt zetaA ρ ≠ ⊤ :=\n    AnalyticOnNhd.analyticOrderAt_ne_top_of_isPreconnected zetaA_analyticOnNhd_univ\n      isPreconnected_univ (Set.mem_univ 1) (Set.mem_univ ρ) (by rw [h0]; simp)\n  have hcong : zetaA =ᶠ[𝓝 ρ] (fun w : ℂ => w - 1) * riemannZeta := by\n    filter_upwards [eventually_ne_nhds hρ] with w hw using zetaA_apply hw\n  rw [analyticOrderAt_congr hcong,\n    analyticOrderAt_mul (f := fun w : ℂ => w - 1) (g := riemannZeta)\n      (analyticAt_id.sub analyticAt_const) (analyticAt_riemannZeta hρ)] at hA\n  have h0' : analyticOrderAt (fun w : ℂ => w - 1) ρ = 0 :=\n    ((show AnalyticAt ℂ (fun w : ℂ => w - 1) ρ from\n      analyticAt_id.sub analyticAt_const).analyticOrderAt_eq_zero).mpr (sub_ne_zero.mpr hρ)\n  rw [h0', zero_add] at hA\n  exact hA"},"extra_content":null},{"type":"tool_use","id":"Edit_36","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","old_string":"/-- (w−1)^{n+1}·ppart n w is the Taylor polynomial of Ffun n at 1 (w ≠ 1). -/\nlemma ppart_mul_pow {n : ℕ} {s : ℂ} (hs : s ≠ 1) :","new_string":"/-- (w−1)^{n+1}·ppart n w is the Taylor polynomial of Ffun n at 1 (w ≠ 1). -/\nlemma ppart_mul_pow (n : ℕ) {s : ℂ} (hs : s ≠ 1) :"},"extra_content":null},{"type":"tool_use","id":"Edit_37","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","old_string":"      have hi' : (s - 1) ^ i ≠ 0 := pow_ne_zero i hs1\n      rw [e]\n      field_simp [hi']\n      ring\n    exact key _","new_string":"      have hi' : (s - 1) ^ i ≠ 0 := pow_ne_zero i hs1\n      rw [e]\n      field_simp [hi']\n    exact key _"},"extra_content":null},{"type":"tool_use","id":"Edit_38","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","old_string":"/-- (w−1)^{n+1}·Jlag n w = Ffun n w off 1 and off the zeros of ζ. -/\nlemma Jlag_mul_pow {n : ℕ} {s : ℂ} (hs1 : s ≠ 1) (hs0 : riemannZeta s ≠ 0) :","new_string":"/-- (w−1)^{n+1}·Jlag n w = Ffun n w off 1 and off the zeros of ζ. -/\nlemma Jlag_mul_pow (n : ℕ) {s : ℂ} (hs1 : s ≠ 1) (hs0 : riemannZeta s ≠ 0) :"},"extra_content":null},{"type":"tool_use","id":"Edit_39","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","old_string":"  rw [hk, hLi_eq_of_ne hs1 hs0, pow_succ]\n  field_simp [h2, hs0]\n  ring","new_string":"  rw [hk, hLi_eq_of_ne hs1 hs0, pow_succ]\n  field_simp [h2, hs0]"},"extra_content":null},{"type":"tool_use","id":"Edit_40","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","old_string":"  have e : (s - 1) ^ (n + 1) * Rfun n s = Ffun n s - TaylP n s := by\n    have e1 := Jlag_mul_pow hs1' hs0\n    have e2 := ppart_mul_pow hs1'","new_string":"  have e : (s - 1) ^ (n + 1) * Rfun n s = Ffun n s - TaylP n s := by\n    have e1 := Jlag_mul_pow n hs1' hs0\n    have e2 := ppart_mul_pow n hs1'"},"extra_content":null}]