[{"type":"thinking","thinking":"Progress — remaining issues:\n\n1. Line 162:88: `(n + 1 - i)!` — STILL the factorial parse issue at Acoeff def. Let me look at line 162: `iteratedDeriv (n + 1 - i) (fun w : ℂ => (w ^ n - (w - 1) ^ n) * hLi w) 1 / (n + 1 - i)!` — the `!` after `)` — \"unexpected token '!'; expected command\". So postfix `!` after a parenthesized expression with space... I have `open scoped Nat` — the notation `n !` needs `n` to be a TERM, and `(n + 1 - i)!` — hmm — maybe the notation requires NO space but also the parser sees `(n + 1 - i)` as complete then `!` as new token — postfix notation should handle this (it's how `n!` fails but `(n+1)!` works?). In Mathlib, `(n + 1)!` is written with no space and works. My text has `(n + 1 - i)!` — should work the same... unless the issue is line 162's def Acoeff — let me look at the exact line. Maybe the earlier `open scoped Nat` was lost in my rewrites? Let me check the file header. Also error 164: the Acoeff_one rw produced `(↑n + 1 - ↑1) = ↑n.factorial` mismatch — `show n + 1 - 1 = n from by omega` rewrote `n + 1 - 1` to `n` in the goal but left `(↑n + 1 - ↑1)` (real cast?) — the factorial argument: Acoeff n 1 = iteratedDeriv (n+1−1) F 1 / (n+1−1)! — rw [show n + 1 - 1 = n] — the target `n + 1 - 1` appears in `iteratedDeriv (n + 1 - 1)` and in `(n + 1 - 1)!` — the display shows the goal became `... / (↑n + 1 - ↑1) = ... / ↑n.factorial` — the `!` in `(n+1-1)!` was elaborated as something weird — `(↑n + 1 - ↑1)` without factorial?! So the factorial parse IS broken at Acoeff's def. Fix: write `Nat.factorial (n + 1 - i)` explicitly.\n\n2. `ContinuousAt.sum` unknown — find the right name. For sums: `ContinuousAt.sum`? — grep needed. Actually there's `ContinuousAt.sum` in Topology.Algebra.Monoid? Hmm — the standard: `ContinuousAt.sum (h : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (∑ i ∈ s, f i) x`? Let me grep.\n\n3. `deriv_sum` usage — `rw [Gprim, deriv_sum (fun i hi => ?_)]` — \"Failed to rewrite using equation theorems for Gprim\" — rw [Gprim] fails to unfold Gprim? Gprim is a def — rw [Gprim] should unfold via equation lemmas — the error \"Failed to rewrite using equation theorems for Gprim\" — maybe Gprim has implicit args or the unfold needs `Gprim.eq_def`? Hmm — rw [Gprim] usually unfolds defs... The issue might be that after the namespace fix, Gprim's def is fine, but rw [Gprim] doesn't unfold because... Let me use `unfold Gprim` or `show` — or `simp only [Gprim]`. Let me use `unfold Gprim` — hmm — or write the rw as `rw [Gprim, deriv_sum ...]` → replace with `unfold Gprim; rw [deriv_sum ...]`. Actually \"Failed to rewrite using equation theorems for Gprim\" typically means the def's equation lemma didn't match the goal's instance — Gprim n w vs the goal's `deriv (Gprim n) w` — the LHS of the equation is `Gprim n w` but the goal has `deriv (Gprim n) w` — the def-eq unfolds Gprim n w to the sum — should match `deriv (Gprim n) w`... Hmm — \"equation theorems for Gprim\" — the error is on `rw [Gprim, deriv_sum ...]` — maybe deriv_sum is the failing one? \"Failed to rewrite using equation theorems for `Gprim`\" — it names Gprim explicitly. Maybe deriv_sum needs the function in lambda form: deriv (fun w => ∑ i, term i w) — after unfolding Gprim, the goal is deriv (∑ i ∈ Icc 2 (n+1), fun w => ...) — deriv_sum's pattern: deriv (∑ i ∈ s, f i) x = ∑ i ∈ s, deriv (f i) x ✓ should match... The error is at line 177 (Gprim_differentiableAt's `rw [Gprim]`) and 187 (deriv_Gprim's `rw [Gprim, deriv_sum]`) — both rw [Gprim] fail. Because Gprim is defined with (n : ℕ) (w : ℂ) explicit args — the equation lemma: Gprim n w = ∑ ... — rw [Gprim] should rewrite Gprim n w... \"Failed to rewrite using equation theorems\" — perhaps because Gprim's definition body has `∑ i ∈ Finset.Icc 2 (n + 1), ...` and the equation lemma generated is `Gprim n w = ...` which should match... Hmm — maybe the issue: `rw [Gprim]` requires the def to be equational/reducible by equation lemmas — for a plain def it generates `Gprim.eq_1`... Actually for non-pattern-matching defs, `rw [Gprim]` uses `Gprim.def`/`Gprim.eq_def`? The standard way: `unfold Gprim` or `simp only [Gprim]`. Let me use `unfold Gprim` — wait — `unfold` works. Or `rw [show Gprim n = fun w => ∑ i ∈ Finset.Icc 2 (n + 1), ... from rfl]`. Use `unfold Gprim` — safe.\n\nHmm wait — actually the reason might be that rw [Gprim] is hitting the ERROR \"Failed to rewrite using equation theorems for `Gprim`\" — this exact error appears when the def has no equation lemmas matching — for defs with explicit arguments, rw [Gprim] needs `Gprim n w` in the goal — we have `deriv (Gprim n) w` — the pattern `Gprim n w` — is `Gprim n w` PRESENT in `deriv (Gprim n) w`? YES as a subterm. Hmm... unless Gprim's first arg n is IMPLICIT in my def? I wrote `def Gprim (n : ℕ) (w : ℂ) : ℂ` — explicit ✓. The application in goal: `Gprim n` applied to w via `deriv (Gprim n) w` — subterm `Gprim n w` ✓ present. Strange. Let me just use `unfold Gprim` — wait, actually maybe the issue is `rw [Gprim]` tries to rewrite with the equation Gprim n w = body, and the body contains `∑ i ∈ Finset.Icc 2 (n + 1), -(Acoeff n i / ((i : ℂ) - 1)) * (w - 1) ^ ...` — no problem... Let me just try `unfold Gprim` and see. Actually — the more likely cause: the previous error (line 162 factorial parse) made Acoeff ill-formed, so Gprim (which mentions Acoeff) is also ill-formed, and rw fails. Fix Acoeff's factorial first — the Gprim errors likely cascade.\n\n4. `ContinuousAt.sum` — check name.\n\n5. tail_continuousAt's hi : ℂ — `intro i hi; rw [Finset.mem_Icc] at hi` — hi got type ℂ?! Because `apply ContinuousAt.sum` failed and hi is garbage. Cascade.\n\n6. hGint's `rw [deriv_Gprim hs1]` — pattern `deriv (Gprim ?m) s` not found in `DifferentiableWithinAt ℂ (deriv (Gprim n)) ... s` — the deriv (Gprim n) UNAPPLIED (as a function) — deriv_Gprim's LHS is `deriv (Gprim n) w` (applied at w) — the goal has the function `deriv (Gprim n)` unapplied at s... it's `DifferentiableWithinAt ℂ (deriv (Gprim n)) (z.Rectangle w \\ {1}) s` — deriv (Gprim n) is a function ℂ → ℂ; applied at s implicitly in DifferentiableWithinAt. The rewrite `deriv (Gprim ?m) ?s` — the goal has `deriv (Gprim n)` as a bare function — the pattern with application doesn't match the unapplied function. Fix: instead of rw, use the fact directly: I want DifferentiableWithinAt (deriv (Gprim n)) ... s — since deriv (Gprim n) = the tail-sum FUNCTION (pointwise equal off {1}), and the tail-sum is continuous at s — actually differentiability of deriv at s: deriv (Gprim n) is differentiable at s (Gprim is analytic off 1 ⟹ deriv analytic) — hmm, the cleaner: show DifferentiableAt ℂ (deriv (Gprim n)) s via: Gprim differentiable on ℂ \\ {1} (open) ⟹ deriv differentiable there (DifferentiableOn.deriv) ⟹ at s ✓:\n```\n    exact ((Gprim_differentiableOn_open (n := n)).deriv (isOpen_compl_singleton)).differentiableWithinAt hs\n```\nHmm — need Gprim_differentiableOn_open : DifferentiableOn ℂ (Gprim n) (Set.univ \\ {1}) or (ℂ \\ {1}): via Gprim_differentiableAt at every w ≠ 1: DifferentiableOn (Gprim n) {1}ᶜ: intro w hw; exact (Gprim_differentiableAt hw).differentiableWithinAt. Then DifferentiableOn.deriv (h : DifferentiableOn ℂ f s) (ho : IsOpen s) : DifferentiableOn ℂ (deriv f) s — with s = {1}ᶜ open ✓. Then .differentiableWithinAt at s ∈ {1}ᶜ ✓ (hs : s ∈ Rectangle \\ {1} — need s ∈ {1}ᶜ: hs.2 ✓).\n\nHmm wait — but HolomorphicOn.rectangleBorderIntegrable' wants HolomorphicOn f (Rectangle \\ {p}) — HolomorphicOn = DifferentiableOn. My goal inside the `intro s hs` is DifferentiableWithinAt ℂ (deriv (Gprim n)) (Rectangle \\ {1}) s. With DifferentiableOn (deriv (Gprim n)) ({1}ᶜ) I get DifferentiableWithinAt at s ∈ {1}ᶜ — but the WithinAt set differs ({1}ᶜ vs Rectangle \\ {1}) — DifferentiableWithinAt f s x with s ⊇ t implies DifferentiableWithinAt f t x: `(h.deriv ...).mono (Set.diff_subset_diff_right (Set.singleton_subset_iff.mpr ?))`... simpler: DifferentiableWithinAt is about the point with a set; from DifferentiableOn f {1}ᶜ at s ∈ {1}ᶜ: `(...).differentiableWithinAt (Set.mem_compl_singleton_iff.mpr hs1)` gives DifferentiableWithinAt f {1}ᶜ s — then `.mono` to Rectangle \\ {1} ⊆ {1}ᶜ: `DifferentiableWithinAt.mono h (Set.diff_subset_diff_left (Set.subset_univ _))`? The set Rectangle \\ {1} ⊆ {1}ᶜ ✓ via diff_subset (Set.subset_univ _). Let me:\n```\n    have hopen : DifferentiableOn ℂ (deriv (Gprim n)) {1}ᶜ :=\n      (Gprim_differentiableOn_compl n).deriv isOpen_compl_singleton\n    exact (hopen s (Set.mem_compl_singleton_iff.mpr hs1)).mono\n      (Set.diff_subset_diff_right (Set.singleton_subset_iff.mpr (Set.mem_univ _)))\n```\nHmm — Set.diff_subset_diff_right (h : t ⊆ s) : a \\ s ⊆ a \\ t — with s := {1}, t := univ?? I want Rectangle \\ {1} ⊆ univ \\ {1}: via Set.diff_subset_diff_right (Set.subset_univ _)?? diff_subset_diff_right (h : s ⊆ t) : a \\ t ⊆ a \\ s — so univ \\ {1} ⊆ ... no I want Rectangle \\ {1} ⊆ univ \\ {1} — a := Rectangle ⊆ univ: Rectangle \\ {1} ⊆ univ \\ {1} via Set.diff_subset_diff_left (Set.subset_univ _) ✓ (diff_subset_diff_left : s ⊆ t → s \\ r ⊆ t \\ r ✓).\n`.mono (Set.diff_subset_diff_left (Set.subset_univ _))` — DifferentiableWithinAt.mono : DifferentiableWithinAt f s x → t ⊆ s → DifferentiableWithinAt f t x ✓ direction: from WithinAt at {1}ᶜ to WithinAt at Rectangle\\{1} ⊆ {1}ᶜ ✓.\n\nHmm wait — actually simpler: `DifferentiableOn.mono`: from DifferentiableOn (deriv Gprim) {1}ᶜ get DifferentiableOn (deriv Gprim) (Rectangle \\ {1}) via .mono (Set.diff_subset_diff_left (Set.subset_univ _)) — then the HolomorphicOn goal is exactly DifferentiableOn (Rectangle \\ {1}) — restructure hGint:\n```\n  have hGint : RectangleBorderIntegrable (deriv (Gprim n)) z w := by\n    apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n    have hopen : DifferentiableOn ℂ (deriv (Gprim n)) {1}ᶜ :=\n      (Gprim_differentiableOn_compl n).deriv isOpen_compl_singleton\n    exact hopen.mono (Set.diff_subset_diff_left (Set.subset_univ _))\n```\nwith Gprim_differentiableOn_compl (n) : DifferentiableOn ℂ (Gprim n) {1}ᶜ:\n```\nlemma Gprim_differentiableOn_compl (n : ℕ) :\n    DifferentiableOn ℂ (Gprim n) {1}ᶜ := by\n  intro w hw\n  exact (Gprim_differentiableAt (Set.mem_compl_singleton_iff.mp hw)).differentiableWithinAt\n```\nAnd `isOpen_compl_singleton : IsOpen {x}ᶜ` ✓ exists.\nAnd `DifferentiableOn.deriv (hd : DifferentiableOn ℂ f s) (ho : IsOpen s) : DifferentiableOn ℂ (deriv f) s` ✓ exists I believe.\n\n7. hAint's HolomorphicOn — similar but simpler (rational function).\n\n8. Also the hAint/hGint rw [RectangleIntegral'_add _ _ hAint hGint] — RectangleIntegral'_add's signature (f g z w) (hf hg) — `RectangleIntegral'_add _ _ hAint hGint` — the _ _ are f g? My RectangleIntegral'_add (f g : ℂ → ℂ) (z w : ℂ) (hf hg) — explicit f g z w — `RectangleIntegral'_add _ _ hAint hGint` gives only 2 underscores (f g) then hAint hGint — but z w also explicit — mismatch! `RectangleIntegral'_add _ _ _ _ hAint hGint`? — hmm — actually looking at my def: `lemma RectangleIntegral'_add (f g : ℂ → ℂ) (z w : ℂ) (hf : ...) (hg : ...)` — so RectangleIntegral'_add _ _ hAint hGint elaborates z w from the hypotheses ✓ — _ _ are f g, then z w from hf/hg ✓ should work. The error at 271:33 says hAint application mismatch — the RectangleIntegral'_add's hf expects RectangleBorderIntegrable (fun w => Acoeff n 1 / (w - 1)) z w ✓ hAint has that ✓ — hmm, the error is truncated; probably cascade.\n\nLet me fix: Acoeff factorial (Nat.factorial explicit), Gprim unfold → `unfold`, ContinuousAt.sum name check, hGint restructure. First check ContinuousAt.sum name.","signature":""},{"type":"text","text":"Checking the remaining API names before fixing."},{"type":"tool_use","id":"Bash_261","name":"Bash","input":{"command":"cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem ContinuousAt.sum\\|lemma ContinuousAt.sum\" Mathlib/ --include=*.lean | head -3; grep -rn \"theorem DifferentiableOn.deriv\\b\\|lemma DifferentiableOn.deriv\\b\" Mathlib/ --include=*.lean | head -3; grep -rn \"theorem isOpen_compl_singleton\" Mathlib/ --include=*.lean | head -2","description":"Check ContinuousAt.sum, DifferentiableOn.deriv, isOpen_compl_singleton"},"extra_content":null}]