[{"type":"thinking","thinking":"Batch of errors:\n\n1. **Line 304 `this` unknown** — in add_nat_add_one_ne_zero_of_mem_ball: my `rw [h] at h1; simp at h1` — wait line 304: let me look. Actually \"Unknown identifier this\" at 304:29 — somewhere I referenced `this`... in digamma_eq_seriesPart_near_zero: `rcases hk0 with habs | hk0; · linarith` — the linarith needs `hmem` etc. — no `this`. Hmm line 304. Let me read the file around 304.\n\n2. **Line 322**: the HasSumUniformlyOn.of_norm_le_summable unification failure: the goal's limit function is `digammaSeriesPart` (a def) but the lemma concludes `fun x => ∑' n, f n x` — not syntactically equal to digammaSeriesPart. Fix: state hu with the lambda limit and convert, or unfold: `have hu : HasSumUniformlyOn (fun j => fun w : ℂ => ...) (fun w => ∑' j, (1/...)) (ball) := by apply ...` then `show`-convert to digammaSeriesPart via defeq... The apply failed because the goal has digammaSeriesPart but the lemma's conclusion has the tsum-lambda — unification up to defeq SHOULD work for apply?? apply uses keyed matching — the head symbol `digammaSeriesPart` vs the lambda — fails. Fix: `rw [show digammaSeriesPart = fun w => ∑' j : ℕ, (1 / ((j:ℂ)+1) - 1/(w + j + 1)) from rfl]` before apply — or state the goal differently: change the hμ statement to use the tsum lambda directly and convert afterwards:\n\n```lean\n  have hu : HasSumUniformlyOn (fun j => fun w : ℂ => 1 / ((j : ℂ) + 1) - 1 / (w + j + 1))\n      (fun w => ∑' j : ℕ, (1 / ((j : ℂ) + 1) - 1 / (w + j + 1))) (Metric.ball 0 (1 / 2)) := by\n    apply HasSumUniformlyOn.of_norm_le_summable hsum\n    ...\n  -- convert to digammaSeriesPart\n  rw [show digammaSeriesPart = fun w => ∑' j : ℕ, (1 / ((j:ℂ)+1) - 1/(w+j+1)) from rfl] at ... \n```\nSimplest: define everything in terms of the lambda and note digammaSeriesPart w = the tsum by rfl — the final DifferentientiableOn goal is about digammaSeriesPart: I'll keep hdiff stated for the lambda and convert at the end:\nActually cleanest: make hu's statement use the tsum-lambda; htend likewise; hdiff : DifferentiableOn ℂ (fun w => ∑' j, ...) ball; then `have : DifferentiableOn ℂ digammaSeriesPart ball := hdiff` (defeq ✓ ascription); then this.analyticOnNhd. Since digammaSeriesPart w ≡ ∑' j, ... by rfl — the FUNCTION equality: digammaSeriesPart = fun w => ∑' ... : rfl? digammaSeriesPart is `def digammaSeriesPart (w : ℂ) : ℂ := ∑' j, ...` — so digammaSeriesPart ≡ fun w => ∑' j, ... by funext-rfl — yes definitional ✓ the ascription works.\n\n3. **Line 353**: `fun (j : ℂ) w` vs `fun (j : ℕ) w` — the hu lambda I wrote `(fun j => fun w : ℂ => ...)` with j inferred as ℂ?! Because the summand `1/((j:ℂ)+1)` — with j : ℕ ascribed inside — hmm the error says hu has type with `j : ℂ` — the lambda binder `fun j => fun w : ℂ => 1 / ((j : ℂ) + 1) - ...` — `(j : ℂ)` coerces j — j's type should be ℕ from the of_norm_le_summable unification (u : ℕ → ℝ)... but the statement of hu as I wrote it pinned nothing — Lean elaborated the binder type from... the error came at the htend line: hu's j got elaborated as ℂ — because in `have hu : HasSumUniformlyOn (fun j => fun w : ℂ => 1 / ((j : ℂ) + 1) - 1 / (w + j + 1)) digammaSeriesPart ...` the ascription `(j : ℂ)` FORCES j : ℂ?? No — `(j : ℂ)` with j of unknown type is a coercion — but then `w + j + 1`: j used without cast in `w + j + 1`?? In my statement I wrote `(w + j + 1)` — with j : ℂ the + is direct. So the elaborator chose j : ℂ (from w + j + 1 needing j : ℂ) and then (j : ℂ) is the identity. The binder type must be pinned: `fun (j : ℕ) => fun w : ℂ => 1 / ((j : ℂ) + 1) - 1 / (w + (j : ℂ) + 1)`. Same class as the journal-028 Finset.image gotcha (pin the lambda type). Fix everywhere the term appears: hu, htend, hdiff's show, the sum.\n\n4. **Line 378**: digamma_eq_seriesPart_near_zero's integerComplement membership: `intro k hk` — goal after intro k: w ∈ integerComplement unfolds to w ∈ (range Int.cast)ᶜ: `intro k hk` — membership in complement of range: ¬∃ k, ↑k = w: intro h where h : ∃ k : ℤ, (k:ℂ) = w: obtain ⟨k, hk⟩ := h with hk : ↑k = w — NOT `intro k hk` (the ∃). My code: `intro k hk` treated it as ∀ k, ... — wrong. The goal after `intro k hk` — error says \"no additional binders\" with k : w ∈ Set.range Int.cast (already the membership proof!) — so the goal was w ∈ (range ...)ᶜ = (range ...) → False: intro k gives k : w ∈ range — then hk fails (no more binders). Fix: \n```lean\n  have hZ : w ∈ Complex.integerComplement := by\n    intro k\n    obtain ⟨k, hk⟩ := k\n    -- hk : (↑k : ℂ) = w\n    ...\n    rw [← hk] at hmem  -- ‖(k:ℂ)‖ < 1/2: wait hmem : ‖w‖ < 1/2: rw [← hk] at hmem: ‖↑k‖ < 1/2 ✓\n```\nHmm — hk : (↑k : ℂ) = w (range membership: ↑k = w). Then hmem : ‖w‖ < 1/2: rw [← hk] at hmem : ‖(k:ℂ)‖ < 1/2 ✓. Then continue as before (hk0 case split etc.), and the final: `exact hw0 (by rw [hk, hk0]; simp)` — hk : ↑k = w, hk0 : k = 0: w = ↑0 = 0: `rw [← hk, hk0]; simp` : goal w = 0: rw [← hk]: ↑k = 0: rw [hk0]: ↑(0:ℤ) = 0: simp ✓.\n\n5. **Line 415**: `AnalyticAt.div analyticAt_const _ (by norm_num : (0:ℂ) − 1 ≠ 0)` — the apply failed: `apply AnalyticAt.div analyticAt_const _ (by norm_num ...)` — the error: \"could not unify AnalyticAt ((fun x => ?m) / HSub.hSub 0) 1 with goal\" — HSub.hSub 0?? The `_` for g got unified weirdly: the hg0 argument `(by norm_num : (0:ℂ) − 1 ≠ 0)` forces g 0 = 0 − 1: g := HSub.hSub 0... i.e. g := fun x => 0 − 1?? Because hg0 : g x ≠ 0 with the ascribed proof about (0:ℂ) − 1: g x =?= 0 − 1 with x := 0: HOU again — g := fun _ => 0−1 or g := HSub.hSub 0 (i.e. fun x => x − 1... wait HSub.hSub 0 is fun x => HSub.hSub 0 x = 0 − x?? no: HSub.hSub 0 : ℂ → ℂ = (0 − ·)?? HSub.hSub a b := a - b so HSub.hSub 0 = fun b => 0 − b — and evaluated at x = 0: 0 − 0 ≠ 0 false... ugh the elaborator picked g := (0 − ·) with x := 1?? since g x = 0 − 1 has solutions g = const (0−1), x anything; or g = (· − 1), x = 0; or g = (0 − ·), x = 1...). The goal g should be fun s => s − 1. Pin it: g := fun s : ℂ => s − 1:\n```lean\n  have hC : AnalyticAt ℂ (fun s : ℂ => 1 / (s - 1)) 0 := by\n    apply AnalyticAt.div analyticAt_const (g := fun s : ℂ => s - 1) _ (by norm_num)\n    exact analyticAt_id.sub analyticAt_const\n```\nHmm — with g pinned: hg0 : (fun s => s − 1) 0 ≠ 0 = (0 − 1 ≠ 0) ✓ norm_num. The remaining goal: AnalyticAt ℂ (fun s => s − 1) 0: `analyticAt_id.sub analyticAt_const` : AnalyticAt ℂ (id − fun _ => 1)?? Pi-sub vs lambda — the exact-defeq risk again: (id − fun _ => 1) 0... as functions: id − const vs fun s => s − 1: Pi.sub_apply defeq ✓ exact should accept. If not: `show AnalyticAt ℂ (id - 1) 0`?? id − 1 as Pi... Let me use a safer construction: `(analyticAt_id (𝕜 := ℂ) (x := (0:ℂ))).sub analyticAt_const`?? same Pi issue. Alternative: fun_prop for AnalyticAt — is there fun_prop support for AnalyticAt? AnalyticAt.deriv was @[fun_prop] — suggesting fun_prop handles AnalyticAt goals. Try `by fun_prop` for these elementary analytic goals. If it works it's much cleaner. Test on hC.\n\n6. **Line 425**: `AnalyticAt.comp (f := fun s : ℂ => s / 2) (digammaSeriesPart_analyticOn 0 ...) h1` — the hg argument: `digammaSeriesPart_analyticOn 0 (Metric.mem_ball_self ...)` : AnalyticAt ℂ digammaSeriesPart 0 — but comp wants AnalyticAt ℂ digammaSeriesPart ((fun s => s/2) 0) = AnalyticAt ℂ digammaSeriesPart (0/2) — 0/2 is not syntactically 0! Fix: convert: have hval : (0:ℂ)/2 = 0 := by norm_num... then the AnalyticAt at 0/2 vs at 0: `hval ▸ ...`? rewrite: \n```lean\n    have hval : (0 : ℂ) / 2 = 0 := by norm_num\n    rw [← hval]\n    exact AnalyticAt.comp (f := fun s : ℂ => s / 2) (digammaSeriesPart_analyticOn 0 (Metric.mem_ball_self (by norm_num))) h1\n```\nwait — the goal is AnalyticAt ℂ (digammaSeriesPart ∘ fun s => s/2) 0 — rw [← hval] rewrites 0 → 0/2 EVERYWHERE including the point 0 in \"∘ ... ) 0\" — breaks. Instead: the comp's result point: AnalyticAt (g ∘ f) x — x stays 0; only the middle needs 0/2 = 0. The issue is only the hg ascription: AnalyticAt.comp expects hg : AnalyticAt ℂ digammaSeriesPart ((fun s => s/2) 0) — defeq to AnalyticAt ℂ digammaSeriesPart 0 (0/2 ≡ 0 ✓ defeq) — but the ERROR says \"expected AnalyticAt ℂ ?m (0/2)\" and my arg has type ...at 0 — defeq should close! The error is an \"Application type mismatch\" from the ELABORATOR not reducing 0/2?? 0/2 for ℂ: HDiv.hDiv 0 2 — div_zero: definitional? 0/2 = 0 * 2⁻¹ = 0 — is that defeq? 0 * 2⁻¹: mul_zero — a REWRITE not defeq! 0/2 does NOT reduce to 0 definitionally in general (0 * c = 0 is mul_zero, propositional for general semirings — actually for ℝ/ℂ, 0 * x = 0 might be defeq via... no, not defeq). So provide hg with the value normalized: \n```lean\n    have hdsp0 : AnalyticAt ℂ digammaSeriesPart (0 / 2 : ℂ) := by\n      rw [show (0 : ℂ) / 2 = 0 from by norm_num]  -- wait (0:ℂ)/2 = 0: div_zero? zero_div: `zero_div : 0 / a = 0` ✓ simp/norm_num\n      exact digammaSeriesPart_analyticOn 0 (Metric.mem_ball_self (by norm_num))\n    exact AnalyticAt.comp (f := fun s : ℂ => s / 2) hdsp0 h1\n```\nwith the rw INSIDE the have's goal: goal AnalyticAt ℂ digammaSeriesPart (0/2): rw [zero_div]: AnalyticAt ... 0 ✓ then exact. `zero_div (2:ℂ)`? rw [zero_div] ✓ (0/a = 0 unconditional ✓).\n\n7. **Line 467**: riemannZeta_neg_two_mul: `rw [h0] at hord` — hord : analyticOrderNatAt riemannZeta (−2↑n) = 1 — h0 : analyticOrderAt ... = 0 — the pattern analyticOrderAt doesn't appear in analyticOrderNatAt (different function!). analyticOrderNatAt f x := (analyticOrderAt f x).toNat presumably — so rw [h0] fails (not unfoldable by rw — need the def unfold: analyticOrderNatAt is a def — `simp only [analyticOrderNatAt]` or `rw [show analyticOrderNatAt riemannZeta (−2*↑n) = (analyticOrderAt riemannZeta (−2*↑n)).toNat from rfl]` then rw [h0]: (0:ℕ∞).toNat = 1: simp: ENat.toNat... `(0 : ℕ∞).toNat = 0`? hmm analyticOrderAt : ℂ → ℕ∞ (WithTop ℕ)? `.toNat` — WithTop.toNat: (0 : WithTop ℕ).toNat = 0 ✓ simp. So:\n```lean\n  have hmz : zeroMult (-2 * (n : ℂ)) = analyticOrderNatAt riemannZeta (-2 * (n : ℂ)) := rfl\n  rw [hmz] at hord  -- hord : analyticOrderNatAt ζ (−2n) = 1\n  rw [show analyticOrderNatAt riemannZeta (-2 * (n : ℂ))\n      = (analyticOrderAt riemannZeta (-2 * (n : ℂ))).toNat from rfl] at hord  -- hmm is analyticOrderNatAt defined as .toNat? check: `analyticOrderNatAt` def: (analyticOrderAt f x).toNat? — the rfl ascription tests it. In LiSides hmz : zeroMult ρ = analyticOrderNatAt ... := rfl and zeroMult := (analyticOrderAt ...).toNat: so analyticOrderNatAt = (analyticOrderAt ...).toNat by the transitivity rfl ✓ — actually zeroMult := (analyticOrderAt ρ).toNat and zeroMult = analyticOrderNatAt by rfl (LiSides) ⟹ analyticOrderNatAt ≡ (analyticOrderAt).toNat definitionally ✓. So:\n  rw [h0] at hord  -- after unfolding? — the unfold: `rw [show analyticOrderNatAt riemannZeta _ = (analyticOrderAt riemannZeta _).toNat from rfl]`?? rw with rfl-proof: `show`-term is rfl: rw [show A = B from rfl] — rw requires a real equation... rfl-proofs work in rw? rw [show A = B from rfl] — it's an Eq — rw accepts but may complain \"motive\"... it rewrites A→B ✓ fine. Simpler: `simp only [analyticOrderNatAt] at hord`?? simp only with a def name uses equation lemmas ✓. Then hord : (analyticOrderAt ζ (−2n)).toNat = 1: rw [h0] at hord ✓ pattern present. Then hord : (0).toNat = 1: simp at hord (WithTop.toNat... (0 : WithTop ℕ).toNat = 0: `WithTop.toNat_zero`? hmm — ENat: analyticOrderAt : ℕ∞ = WithTop ℕ: .toNat: `WithTop.toNat`? — simp should know (0:ℕ∞).toNat = 0 (ENat.toNat_zero? or via coe). Then hord : 0 = 1 — simp closes (contradiction) or `exact zero_ne_one hord` / norm_num at hord ✓.\n```\nLet me also double check `analyticOrderNatAt` def to confirm it's `.toNat`:","signature":""},{"type":"text","text":"Checking `analyticOrderNatAt`'s definition and then applying the batch of fixes."},{"type":"tool_use","id":"Bash_144","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"def analyticOrderNatAt\\|noncomputable def analyticOrderNatAt\" Mathlib/Analysis/Analytic/ -r | head -2 && grep -rn -A 2 \"def analyticOrderNatAt\" Mathlib/Analysis/Analytic/Order.lean | head -5; grep -rn \"toNat_zero\\|toNat_eq_zero\" Mathlib/Data/ENat/Basic.lean Mathlib/Data/ENat/*.lean 2>/dev/null | head -4","description":"analyticOrderNatAt def and ENat toNat_zero"},"extra_content":null}]