[{"type":"thinking","thinking":"STILL mis-unifying: h3 : AnalyticAt ℂ (digamma ∘ HDiv.hDiv (1 + 0)) 2 — h1 : AnalyticAt ℂ digamma ((fun s => (1+s)/2) 0) — the context shows h1 with the application form ✓ — but the comp still produced digamma ∘ HDiv.hDiv (1+0)!! — so the unifier read h1's point `(fun s => (1+s)/2) 0` and decomposed it as (HDiv.hDiv (1+0)) applied to 2?!?! — WAIT — `(fun s => (1+s)/2) 0` — if Lean DOESN'T beta-reduce during the match, it sees `Function.app (fun s => (1+s)/2) 0` — and matching against `(f ?x)`: f := fun s => (1+s)/2, x := 0 ✓ should work. But it produced digamma ∘ HDiv.hDiv (1+0) — meaning it beta-reduced h1's point to (1+0)/2 = HDiv.hDiv (1+0) 2 and THEN matched f := HDiv.hDiv (1+0), x := 2. So the elaborator DID beta-reduce the point in h1's type (h1's stated type's point was beta-reduced during elaboration!). The beta-reduction during elaboration of the have's type — Lean elaborates the type and WHNF-normalizes it? — for the ascription it might. So keeping the application unreduced via the type ascription doesn't survive.\n\nTo defeat this: keep the point as a VARIABLE-free form that can't beta-reduce... — OR use AnalyticAt.comp_of_eq (the separate-equality version) which avoids the unification entirely:\n```lean\n      have h1 : AnalyticAt ℂ Complex.digamma ((1 : ℂ) / 2) :=\n        digamma_analyticOn _ dpsiDomain_half\n      have h2 : AnalyticAt ℂ (fun s : ℂ => (1 + s) / 2) 0 :=\n        (analyticAt_const.add analyticAt_id).div analyticAt_const (by norm_num)\n      have h3 := AnalyticAt.comp_of_eq h1 h2 (by simp)\n```\ncomp_of_eq (hg : AnalyticAt g y) (hf : AnalyticAt f x) (hy : f x = y) — with g := digamma, y := 1/2, f := the affine, x := 0 — hy : (fun s => (1+s)/2) 0 = 1/2 — by simp — the simp normalizes the application to (1+0)/2 and then to 1/2 ✓ — the y here is EXPLICIT in the hypothesis hy's type — no bad unification (y := 1/2 from h1's point ✓ and f x := (fun s => (1+s)/2) 0 — beta — the hy's type after elaboration: (fun s => (1+s)/2) 0 = 1/2 — hmm — the (fun s => (1+s)/2) 0 — is it beta-reduced in hy's elaborated type? — the hy's expected type: f x = y — with f := affine, x := 0, y := 1/2 — so hy : (fun s => (1+s)/2) 0 = 1/2 — elaborated — the LHS might beta-reduce to (1+0)/2 — and then the proof `by simp` closes ((1+0)/2 = 1/2) ✓ either way ✓ — the point is comp_of_eq does the ▸ internally with hy — no unification of (f x) against the point ✓✓.\n\nLet me use comp_of_eq. Similarly the ContinuousAt hG.comp — same issue at line 156 — ContinuousAt.comp (hf : ContinuousAt f (g x)) (hg) — same problem — is there ContinuousAt.comp_of_eq? — let me check... probably not with that name. Alternative for the ContinuousAt comp: use `ContinuousAt.comp` with hG's point as the application — same beta problem. Hmm — the ContinuousAt version: `ContinuousAt.comp (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x)` — I need a comp_of_eq for ContinuousAt — check quickly. If missing, do the transport manually:\n```lean\n    have hG0 : ContinuousAt Complex.Gamma ((1 : ℂ) / 2) := [the pole-free proof]\n    have hcomp : ContinuousAt (fun s : ℂ => Complex.Gamma ((1 + s) / 2)) 0 := by\n      have he : ((fun s : ℂ => (1 + s) / 2)) 0 = (1 : ℂ) / 2 := by simp\n      -- ContinuousAt.comp with the point as application — or directly via tendsto:\n      have h := hG0.comp_of_tendsto?? \n```\nHmm — alternatively avoid ContinuousAt entirely — I only need the Tendsto.eventually_ne for the Gamma ≠ 0: use hcont1 := ContinuousAt (fun s => Gamma ((1+s)/2)) 0 — build via: the composition Gamma ∘ affine is continuous at 0 because both are continuous — `ContinuousAt.comp`... — the point unification issue — let me use the nhds-level approach: Tendsto (fun s => Gamma((1+s)/2)) (𝓝 0) (𝓝 (Gamma (1/2))) — via `Tendsto.comp`: Tendsto (affine) (𝓝 0) (𝓝 (affine 0)) [the affine's continuity at 0: ContinuousAt.tendsto] and Tendsto Gamma (𝓝 (affine 0)) (𝓝 (Gamma (affine 0))) [Gamma's continuity at affine 0 = 1/2] — the comp: Tendsto (Gamma ∘ affine) (𝓝 0) (𝓝 (Gamma (affine 0))) — with affine 0 = (1+0)/2 — hmm — the same beta thing but at the Tendsto level — `Tendsto.comp (hg : Tendsto g (𝓝 (f x)) ...)`? — hmm.\n\nLet me look for ContinuousAt.comp_of_eq / Tendsto comp forms.","signature":""},{"type":"text","text":"The elaborator keeps beta-reducing the point and mis-decomposing it. Let me use the `_of_eq` variants — checking for the ContinuousAt one."},{"type":"tool_use","id":"Bash_580","name":"Bash","input":{"command":"cd \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib\" && grep -rn \"theorem ContinuousAt.comp_of_eq\\|lemma ContinuousAt.comp_of_eq\\|ContinuousAt.comp_of_eq\" Mathlib/Topology/ -r --include=\"*.lean\" | head -3; grep -rn \"theorem ContinuousAt.comp\\b\" Mathlib/Topology/Algebra/GroupWithZero.lean Mathlib/Topology/Continuous*.lean 2>/dev/null | head -3","description":"Check ContinuousAt comp variants"},"extra_content":null}]