[{"type":"thinking","thinking":"Line 389 col 54: `have hcpi : AnalyticAt ℂ (fun s => (Real.pi : ℂ) ^ (-s / 2)) (-2) := by` — col 54 is around `^ (-s / 2)` — the synth failures `Neg ℕ` at 389:54/64 and `NormedAddCommGroup ℕ` at 389:14. Hmm — 389:14 is `AnalyticAt ℂ (fun s => ...` — wait: `NormedAddCommGroup ℕ`?? The `(-2)` at end: `AnalyticAt ℂ f (-2)` — could elaborate the point as... AnalyticAt 𝕜 f z₀ with f : ℂ → ℂ, z₀ : ℂ ✓.\n\n`Neg ℕ` at 389:54 and 389:64 — the `(-s / 2)` has TWO negations?? No, one `-s`. Col 54/64: `(Real.pi : ℂ) ^ (-s / 2)` — hmm, `(-s / 2)` — `-s` needs Neg ℂ ✓ fine...\n\nWait — maybe it's the cpow: `↑Real.pi ^ (-s/2)` — which `^`? With base ℂ and exponent ℂ → cpow ✓. But what if `-s/2` elaborated weirdly: `-s / 2` with s : ℂ ✓.\n\nHmm, actually — 389:54 — let me count precisely: `  have hcpi : AnalyticAt ℂ (fun s => (Real.pi : ℂ) ^ (-s / 2)) (-2) := by`. Col 54 ≈ `(Real.pi : ℂ)`. Col 64 ≈ `^ (-s / 2)`. The `Neg ℕ` — where's an ℕ?? OH — I bet it's `(Real.pi : ℂ)` — no wait.\n\nHmm, `NormedAddCommGroup ℕ` at 389:14 — AnalyticAt ℂ (fun s => ...) needs the codomain to be a NormedAddCommGroup... if the lambda's body got elaborated with s : ℕ?! The lambda `fun s => (Real.pi : ℂ) ^ (-s / 2)` — s's type is unbound! In `AnalyticAt ℂ (fun s => ...)`, AnalyticAt 𝕜 f z₀ with f : 𝕜 → E — the elaborator must infer s : ℂ from unification with AnalyticAt ℂ ?f ?z₀... but the expected type of the `have` — I wrote `AnalyticAt ℂ (fun s => (Real.pi : ℂ) ^ (-s / 2)) (-2)` — the lambda binds s without type annotation, and elaboration should unify ?f : ℂ → ?E forcing s : ℂ. But apparently it defaulted to ℕ?? (defaulting rules: unassigned type metavariable for the binder defaults to... ℕ!). Right — Lean4 defaulting: `fun s => ...` with unknown binder type defaults to ℕ in some situations. But normally AnalyticAt ℂ (fun s => ...) unifies f's domain with ℂ first... apparently not here (maybe because the `have` type is elaborated outside-in and the lambda is elaborated before the AnalyticAt application unifies).\n\nFix: annotate the binder: `(fun s : ℂ => (Real.pi : ℂ) ^ (-s / 2))`. Same for all bare `fun s =>` in types: hGamma's has `fun s => Complex.Gamma (s / 2 + 2)` — s/2 forces ℂ? `s / 2` with s : ℕ would... /2 on ℕ is ℕ division — and then Gamma (ℕ thing) needs ℂ coercion... The earlier error 414 shows `DifferentiableAt ℂ (Gamma ∘ fun s => s / 2 + 2) s` type mismatch with expected `DifferentiableWithinAt ℂ (fun s => Gamma (s / 2 + 2)) (ball ...) s` — the goal inside DifferentiableOn is DifferentiableWithinAt! I forgot: `DifferentiableOn ℂ f s` unfolds to ∀ x ∈ s, DifferentiableWithinAt ℂ f s x — intro s hs gives DifferentiableWithinAt goal, not DifferentiableAt! Need `.differentiableWithinAt` at the end: (hg.comp ...).differentiableWithinAt ✓.\n\nSo fixes:\n1. hcpi: annotate `(fun s : ℂ => ...)`.\n2. hGamma's hd: end with `(hg.comp s (...)).differentiableWithinAt`... wait — goal after intro s hs: DifferentiableWithinAt ℂ (fun s => Gamma (s/2+2)) (ball) s. Provide: ((hg.comp s (show ...)).differentiableWithinAt : DifferentiableWithinAt ℂ (Gamma ∘ (fun s => s/2+2)) (ball) s — the ∘ form vs fun s => Gamma (s/2+2): defeq ✓ accepted by exact.\n\nHmm but the error 414: \"Type mismatch DifferentiableAt.comp s hg (this) has type DifferentiableAt ℂ (Gamma ∘ fun s => s/2+2) s but expected DifferentiableWithinAt ℂ (fun s => Gamma (s/2+2)) (ball) s\" — so .differentiableWithinAt fixes it ✓.\n\n3. The hEv/zeta eventuallyEq might have similar fun-binder issues — the filter_upwards goal will have s ascribed from context ✓ fine.\n\n4. Line 421's `NormedAddCommGroup ℕ` — the `show AnalyticAt ℂ (fun s => ...) (−2)` — same bare-lambda defaulting! Annotate `(fun s : ℂ => ...)`.\n\n5. Also hz0's `simp` at −2 case and other spots probably fine.\n\n6. The `apply hexp.congr` failure — could not unify because the lambda types were wrong (ℕ defaulting). With (fun s : ℂ => ...) annotation, `hexp.congr` should unify ?m with the cpow function. But wait — apply ... congr needs f =ᶠ[𝓝 x] g with f = the exp fn, g = the cpow fn: after annotation it should unify. Keep `apply hexp.congr; filter_upwards ...`. Hmm — but the filter_upwards goal direction: f =ᶠ g means ∀ᶠ s, f s = g s: goal `Complex.exp (log π * (−1/2) * s) = ↑Real.pi ^ (−s / 2)` — then rw [cpow_def_of_ne_zero] rewrites RHS ✓ then ring_nf for exp args ✓.\n\nWait — also `(analyticAt_const.mul analyticAt_id).cexp` — analyticAt_const : AnalyticAt ℂ (fun _ => v) x ✓; .mul analyticAt_id : AnalyticAt ℂ ((fun _ => v) * id)?? AnalyticAt.mul gives (fun s => v * s)? AnalyticAt.mul (hf : AnalyticAt f) (hg : AnalyticAt g) : AnalyticAt (f * g) where f * g is pointwise: (f*g) s = v * s ✓ and hexp's type ascription (fun s => exp ((log π * (−1/2)) * s)) — the ascription drives unification ✓ (with v := log π * (−1/2)). The .cexp: AnalyticAt.cexp (fa : AnalyticAt ℂ f x) : AnalyticAt ℂ (exp ∘ f) x — target fun s => exp(c * s) vs exp ∘ (fun s => c*s) — defeq, the ascription handles ✓.\n\nAlso `(by norm_num : (Real.pi : ℂ) ≠ 0)` ✓.\n\nLet me apply: annotate the binders + differentiableWithinAt fix.","signature":""},{"type":"text","text":"Classic elaboration issue: bare `fun s =>` binders in the type ascriptions default the domain to ℕ. Annotating binders + the `DifferentiableWithinAt` fix (DifferentiableOn's goal form)."},{"type":"tool_use","id":"Edit_119","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean","old_string":"theorem Hfun_analyticAt_neg_two : AnalyticAt ℂ Hfun (-2) := by\n  have hcpi : AnalyticAt ℂ (fun s => (Real.pi : ℂ) ^ (-s / 2)) (-2) := by\n    have hexp : AnalyticAt ℂ\n        (fun s => Complex.exp ((Complex.log (Real.pi : ℂ) * (-1 / 2)) * s)) (-2) :=\n      (analyticAt_const.mul analyticAt_id).cexp\n    apply hexp.congr\n    filter_upwards with s\n    rw [Complex.cpow_def_of_ne_zero (by norm_num : (Real.pi : ℂ) ≠ 0)]\n    ring_nf\n  have hGamma : AnalyticAt ℂ (fun s => Complex.Gamma (s / 2 + 2)) (-2) := by\n    have hd : DifferentiableOn ℂ (fun s => Complex.Gamma (s / 2 + 2))\n        (Metric.ball (-2 : ℂ) (1 : ℝ)) := by\n      intro s hs\n      have hg : DifferentiableAt ℂ Complex.Gamma (s / 2 + 2) := by\n        apply Complex.differentiableAt_Gamma\n        intro m hm\n        have hs_eq : s = -2 * (m : ℂ) - 4 := by linear_combination 2 * hm\n        rw [hs_eq, Metric.mem_ball, dist_eq_norm] at hs\n        have h1 : ((-2 : ℂ) * (m : ℂ) - 4) - -2\n            = -(((2 : ℝ) * m + 2 : ℝ) : ℂ) := by push_cast; ring\n        rw [h1, norm_neg, Complex.norm_real, Real.norm_eq_abs] at hs\n        rw [abs_of_nonneg (by positivity : (0:ℝ) ≤ 2 * (m : ℝ) + 2)] at hs\n        have hge : (2:ℝ) ≤ 2 * (m : ℝ) + 2 := by\n          have hm0 : (0:ℝ) ≤ (m : ℝ) := by positivity\n          linarith\n        linarith\n      exact hg.comp s\n        (show DifferentiableAt ℂ (fun s : ℂ => s / 2 + 2) s from by fun_prop)\n    have hball : (-2 : ℂ) ∈ Metric.ball (-2 : ℂ) (1 : ℝ) :=\n      Metric.mem_ball_self (zero_lt_one : (0:ℝ) < 1)\n    exact (hd.analyticOnNhd Metric.isOpen_ball) _ hball\n  show AnalyticAt ℂ (fun s => 4 * (Real.pi : ℂ) ^ (-s / 2) *\n    Complex.Gamma (s / 2 + 2) / s) (-2)\n  exact ((analyticAt_const.mul hcpi).mul hGamma).div analyticAt_id (by norm_num)","new_string":"theorem Hfun_analyticAt_neg_two : AnalyticAt ℂ Hfun (-2) := by\n  have hcpi : AnalyticAt ℂ (fun s : ℂ => (Real.pi : ℂ) ^ (-s / 2)) (-2) := by\n    have hexp : AnalyticAt ℂ\n        (fun s : ℂ => Complex.exp ((Complex.log (Real.pi : ℂ) * (-1 / 2)) * s)) (-2) :=\n      (analyticAt_const.mul analyticAt_id).cexp\n    apply hexp.congr\n    filter_upwards with s\n    rw [Complex.cpow_def_of_ne_zero (by norm_num : (Real.pi : ℂ) ≠ 0)]\n    ring_nf\n  have hGamma : AnalyticAt ℂ (fun s : ℂ => Complex.Gamma (s / 2 + 2)) (-2) := by\n    have hd : DifferentiableOn ℂ (fun s : ℂ => Complex.Gamma (s / 2 + 2))\n        (Metric.ball (-2 : ℂ) (1 : ℝ)) := by\n      intro s hs\n      have hg : DifferentiableAt ℂ Complex.Gamma (s / 2 + 2) := by\n        apply Complex.differentiableAt_Gamma\n        intro m hm\n        have hs_eq : s = -2 * (m : ℂ) - 4 := by linear_combination 2 * hm\n        rw [hs_eq, Metric.mem_ball, dist_eq_norm] at hs\n        have h1 : ((-2 : ℂ) * (m : ℂ) - 4) - -2\n            = -(((2 : ℝ) * m + 2 : ℝ) : ℂ) := by push_cast; ring\n        rw [h1, norm_neg, Complex.norm_real, Real.norm_eq_abs] at hs\n        rw [abs_of_nonneg (by positivity : (0:ℝ) ≤ 2 * (m : ℝ) + 2)] at hs\n        have hge : (2:ℝ) ≤ 2 * (m : ℝ) + 2 := by\n          have hm0 : (0:ℝ) ≤ (m : ℝ) := by positivity\n          linarith\n        linarith\n      exact (hg.comp s\n        (show DifferentiableAt ℂ (fun s : ℂ => s / 2 + 2) s from by fun_prop))\n          .differentiableWithinAt\n    have hball : (-2 : ℂ) ∈ Metric.ball (-2 : ℂ) (1 : ℝ) :=\n      Metric.mem_ball_self (zero_lt_one : (0:ℝ) < 1)\n    exact (hd.analyticOnNhd Metric.isOpen_ball) _ hball\n  show AnalyticAt ℂ (fun s : ℂ => 4 * (Real.pi : ℂ) ^ (-s / 2) *\n    Complex.Gamma (s / 2 + 2) / s) (-2)\n  exact ((analyticAt_const.mul hcpi).mul hGamma).div analyticAt_id (by norm_num)"},"extra_content":null}]