← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (7712 chars)
Issues:
1. **Line 155**: `h.div_const (2 : ℂ)` — the error "numerals are data... ContinuousAt (HDiv.hDiv (1+0)) 2 : Prop" — the div_const's elaboration — `h.div_const (2:ℂ)` — ContinuousAt.div_const (hf : ContinuousAt f x) (c : 𝕜)? — hmm — the error suggests div_const isn't matching. Let me check ContinuousAt.div_const's signature — maybe it's `ContinuousAt.div_const (hf : ContinuousAt f x) (c : 𝕜) : ContinuousAt (fun y => f y / c) x`? — the error "ContinuousAt (HDiv.hDiv (1+0)) 2" — the elaborator parsed `h.div_const (2:ℂ)` as... hmm — maybe the method is different. Let me check what `.div_const` resolves to for ContinuousAt.
Actually — hold on, the error at 155 is in hcont1's haffine: `simpa using h.div_const (2 : ℂ)` — and the error shows `ContinuousAt (HDiv.hDiv (1 + 0)) 2 : Prop` — the div_const application elaborated h.div_const's argument... — let me check if ContinuousAt.div_const exists. If not, use: `(continuousAt_const.add continuousAt_id).div continuousAt_const two_ne_zero`? — the ContinuousAt.div needs the side — `ContinuousAt.div (hf) (hg) (h : g x ≠ 0)`? — the side (fun _ => 2) 0 ≠ 0 — `by norm_num`... — earlier that errored differently. Let me check the names quickly.
2. **Line 218**: h1'.comp h2 — the h1' : AnalyticAt ℂ digamma ((fun s => (1+s)/2) 0) — and h2 : AnalyticAt ℂ (fun s => (1+s)/2) 0 — the comp: AnalyticAt.comp (hf : AnalyticAt f (g x)) (hg : AnalyticAt g x) — with g := fun s => (1+s)/2, x := 0: g x = (fun s => (1+s)/2) 0 — and h1's point must be exactly that application form `(fun s => (1+s)/2) 0` — my h1' HAS that form ✓ — but the error says h2 "is expected to have type AnalyticAt ℂ (HDiv.hDiv (1+0)) 2" — so the elaborator unified g := HDiv.hDiv (1+0)?!?! — because h1''s point `(fun s => (1+s)/2) 0` — Lean read the APPLICATION form and tried to decompose it as (g 0) with g := HDiv.hDiv (1+0) (treating the application as (something) 0!) — since `(fun s => (1+s)/2) 0` is an application of (fun s => ...) to 0 — Lean's unifier for h1'.comp h2 needs (g ?x) with ?x — the point in h1' is the raw application (fun s => (1+s)/2) 0 — the unifier might beta-reduce it to (1+0)/2 then fail to match against (g 0) with g := fun s => (1+s)/2 — should be FINE (beta)... — the error suggests the unification assigned g := HDiv.hDiv (1+0) — i.e., it parsed h1''s point as `HDiv.hDiv (1+0) 2`?? — no wait — the expected type for h2 is `AnalyticAt ℂ (HDiv.hDiv (1+0)) 2` — that's AnalyticAt of the function `HDiv.hDiv (1+0)` at point `2`!! — the unifier saw h1''s point as (fun s => (1+s)/2) 0 and decomposed... no. It decomposed h1''s point `((fun s => (1+s)/2)) 0` — hmm — the expected type of h2 is AnalyticAt ℂ (HDiv.hDiv (1+0)) 2 — so the unifier set g := (HDiv.hDiv (1+0))?? and x := 2 — i.e., it read h1''s point as `(HDiv.hDiv (1+0)) 2`?? — that means h1''s point elaborated to `HDiv.hDiv (1+0) 2` — WAIT — h1''s point I wrote as `((fun s : ℂ => (1 + s) / 2) 0)` — that's ((fun s => (1+s)/2)) applied to 0 — beta — (1+0)/2 — hmm — but the ERROR says the expected h2 type has point 2 and function HDiv.hDiv (1+0) — so Lean parsed h1''s point `((fun s : ℂ => (1 + s) / 2) 0)` as `HDiv.hDiv (1+0) 2`?? — no — `(fun s => (1+s)/2) 0` — how would that become HDiv.hDiv (1+0) 2... — `(1+s)/2` with s := 0: (1+0)/2 = HDiv.hDiv (1+0) 2 — YES — that's the beta-REDUCED form — and h1' has that as its point — so h1' : AnalyticAt ℂ digamma (HDiv.hDiv (1+0) 2) — i.e. digamma ((1+0)/2) — and the comp's unification: AnalyticAt.comp wants hf : AnalyticAt f (g x) — with h1''s point being (1+0)/2 = HDiv.hDiv (1+0) 2 — the unifier needs to see it as (g x) — it chose g := HDiv.hDiv (1+0), x := 2?!?! — because `(1+0)/2` as `(g x)` with g := `HDiv.hDiv (1+0)` and x := 2 — YES that's a valid first-order match (the term (1+0)/2 IS HDiv.hDiv (1+0) 2 — with the last-arg-as-application pattern)!! The unifier prefers that wrong match. To force the right one: state h1''s point as literally `(fun s : ℂ => (1+s)/2) 0` UNREDUCED — I did that with the simpa — but simpa normalized it to (1+0)/2 = HDiv.hDiv (1+0) 2 — the beta-reduced form — which then mis-unifies. Fix: DON'T let it normalize — use `show`:
```lean
have h1' : AnalyticAt ℂ Complex.digamma ((fun s : ℂ => (1 + s) / 2) 0) := by
show AnalyticAt ℂ Complex.digamma ((1 + (0:ℂ))/2) -- beta of the application — hmm the point is the application (fun s => (1+s)/2) 0 — and I show (1+0)/2 — these are defeq — the SHOW changes h1''s type to the (1+0)/2 form — same problem.
```
The problem: I NEED h1''s point to be the UNREDUCED application `(fun s => (1+s)/2) 0` for the comp's unifier to pick g := fun s => (1+s)/2. But any tactic normalizes it. Alternative: avoid the issue by using `AnalyticAt.comp'`? or by specifying g via the comp's implicit args: `AnalyticAt.comp (f := Complex.digamma) (g := fun s : ℂ => (1+s)/2) h1 h2`? — if the f/g are implicit named args I can pin them:
```lean
exact AnalyticAt.comp (g := fun s : ℂ => (1 + s) / 2) h1 h2 |>.contDiffAt
```
hmm — AnalyticAt.comp's explicit args are (hf) (hg) — the f, g, x implicit — with g pinned, the unification of h1's point (1/2 — via my h1's direct form) with (g 0) = (fun s => (1+s)/2) 0 — beta to (1+0)/2 — vs h1's point 1/2 — defeq ✓ — the explicit g-pinning should force the right match. Let me use:
```lean
exact (AnalyticAt.comp (g := fun s : ℂ => (1 + s) / 2) h1 h2).contDiffAt
```
with h1 : AnalyticAt ℂ Complex.digamma (1/2) directly (no h1' detour) — the (g 0) = (fun s => (1+s)/2) 0 — beta → (1+0)/2 — vs h1's 1/2 — defeq ✓ — hmm — will the elaborator reduce (fun s => (1+s)/2) 0 to (1+0)/2 then to 1/2? — for the defeq check it reduces ✓ should pass. And the resulting type: AnalyticAt ℂ (digamma ∘ (fun s => (1+s)/2)) 0 — then the heq bridge to the lambda ✓ then .contDiffAt ✓.
3. **Line 224**: `hcd0.const_mul _` — ContDiffAt.const_mul — "Function.const_mul does not have a usable parameter" — wrong namespace resolution — the method: `ContDiffAt.const_mul (c) (h : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (fun y => c * f y) x` — dot-form h.const_mul c? — or the const_mul has args (c) then the ContDiffAt — `hcd0.const_mul` — hcd0 : ContDiffAt — hcd0.const_mul — should resolve ContDiffAt.const_mul ✓ — the error says Function.const_mul — the elaborator found Function.const_mul (a different thing!) — because ContDiffAt.const_mul doesn't exist with that name? — the iteratedDeriv_const_mul exists (used in LiResidue) — for ContDiffAt: `ContDiffAt.const_mul`? — hmm — maybe it's `ContDiffAt.const_mul (c : 𝔸) (hf : ContDiffAt 𝕜 n f x)` — the arg order (c first) — so `hcd0.const_mul _` would still work via dot... the error shows Function.const_mul — meaning ContDiffAt.const_mul wasn't found and it fell to Function.const_mul?? no — dot-notation for hcd0 : ContDiffAt — resolves ContDiffAt.const_mul first — if missing, error "unknown constant ContDiffAt.const_mul" — the error instead tried Function.const_mul — bizarre. Maybe the name is `ContDiffAt.const_mul` — let me check. Or use `ContDiffAt.mul (contDiff_const.contDiffAt) hcd0`? — that gives (const * f) — the Pi.mul form — my target (fun s => (1/2) * digamma(...)) — the lambda-of-product — ContDiffAt.mul : ContDiffAt (f * g) — Pi.mul — vs the lambda — defeq via Pi.mul_apply ✓. Use:
```lean
have hcd : ContDiffAt ℂ j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=
(contDiff_const (𝕜 := ℂ)).contDiffAt.mul hcd0
```
hmm — contDiff_const : ContDiff 𝕜 n fun _ => c — .contDiffAt : ContDiffAt — then .mul hcd0 : ContDiffAt ((fun _ => c) * (digamma-comp)) 0 — Pi.mul — the target lambda — defeq ✓. But the c's inference — `contDiff_const` needs the c — from the target's ascription ✓ (hcd's stated type pins it) ✓.
Let me apply these.
Fixing the comp unification (pin g) and the const_mul resolution.
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean",
"old_string": " have hcd0 : ContDiffAt \u2102 j (fun s => Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt \u2102 Complex.digamma ((1 : \u2102) / 2) :=\n digamma_analyticOn _ dpsiDomain_half\n have h2 : AnalyticAt \u2102 (fun s : \u2102 => (1 + s) / 2) 0 :=\n (analyticAt_const.add analyticAt_id).div analyticAt_const (by norm_num)\n have h1' : AnalyticAt \u2102 Complex.digamma ((fun s : \u2102 => (1 + s) / 2) 0) := by\n simpa using h1\n have h3 := h1'.comp h2\n have heq : (Complex.digamma \u2218 (fun s : \u2102 => (1 + s) / 2)) =\n fun s => Complex.digamma ((1 + s) / 2) := rfl\n rw [heq] at h3\n exact h3.contDiffAt\n have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=\n hcd0.const_mul _",
"new_string": " have hcd0 : ContDiffAt \u2102 j (fun s => Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt \u2102 Complex.digamma ((1 : \u2102) / 2) :=\n digamma_analyticOn _ dpsiDomain_half\n have h2 : AnalyticAt \u2102 (fun s : \u2102 => (1 + s) / 2) 0 :=\n (analyticAt_const.add analyticAt_id).div analyticAt_const (by norm_num)\n have h3 := AnalyticAt.comp (g := fun s : \u2102 => (1 + s) / 2) h1 h2\n have heq : (Complex.digamma \u2218 (fun s : \u2102 => (1 + s) / 2)) =\n fun s => Complex.digamma ((1 + s) / 2) := rfl\n rw [heq] at h3\n exact h3.contDiffAt\n have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=\n (contDiff_const (\ud835\udd5c := \u2102) (c := -(Real.log Real.pi : \u2102) / 2 * 0 + (1 / 2))).contDiffAt.mul hcd0"
}
result 2026-08-22 11:09:39Z → 2026-08-22 11:09:39Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean",
"oldString": " have hcd0 : ContDiffAt \u2102 j (fun s => Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt \u2102 Complex.digamma ((1 : \u2102) / 2) :=\n digamma_analyticOn _ dpsiDomain_half\n have h2 : AnalyticAt \u2102 (fun s : \u2102 => (1 + s) / 2) 0 :=\n (analyticAt_const.add analyticAt_id).div analyticAt_const (by norm_num)\n have h1' : AnalyticAt \u2102 Complex.digamma ((fun s : \u2102 => (1 + s) / 2) 0) := by\n simpa using h1\n have h3 := h1'.comp h2\n have heq : (Complex.digamma \u2218 (fun s : \u2102 => (1 + s) / 2)) =\n fun s => Complex.digamma ((1 + s) / 2) := rfl\n rw [heq] at h3\n exact h3.contDiffAt\n have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=\n hcd0.const_mul _",
"newString": " have hcd0 : ContDiffAt \u2102 j (fun s => Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt \u2102 Complex.digamma ((1 : \u2102) / 2) :=\n digamma_analyticOn _ dpsiDomain_half\n have h2 : AnalyticAt \u2102 (fun s : \u2102 => (1 + s) / 2) 0 :=\n (analyticAt_const.add analyticAt_id).div analyticAt_const (by norm_num)\n have h3 := AnalyticAt.comp (g := fun s : \u2102 => (1 + s) / 2) h1 h2\n have heq : (Complex.digamma \u2218 (fun s : \u2102 => (1 + s) / 2)) =\n fun s => Complex.digamma ((1 + s) / 2) := rfl\n rw [heq] at h3\n exact h3.contDiffAt\n have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=\n (contDiff_const (\ud835\udd5c := \u2102) (c := -(Real.log Real.pi : \u2102) / 2 * 0 + (1 / 2))).contDiffAt.mul hcd0",
"originalFile": "/-\nNguyenRH1/LiArchimedean.lean \u2014 tier ii\u2032 sub-stage ii\u2032-3a, piece (A)\n(D-066 \u00a74): the archimedean coefficients of the (4.10) split.\n\nTarget (D-066): the Taylor coefficients of s \u21a6 logDeriv Gamma\u211d (1+s) at 0:\n\n (1/j!)\u00b7iteratedDeriv j (fun s => logDeriv Gamma\u211d (1+s)) 0 = (\u22121)^{j+1} a_{j+1},\n\ni.e. `iteratedDeriv j ... 0 = (\u22121)^{j+1}\u00b7j!\u00b7lagariasA (j+1)`.\n\nPart 1 (this file, complete): the Gamma\u211d logDeriv split\n logDeriv Gamma\u211d (1+s) = \u2212(log \u03c0)/2 + (1/2)\u00b7digamma((1+s)/2)\n(`logDeriv_Gamma\u211d_one_add`) and the j = 0 evaluation\n logDeriv Gamma\u211d 1 = \u2212(log 4\u03c0 + \u03b3)/2 = \u2212lagariasA1\n(`logDeriv_Gamma\u211d_one`) \u2014 via Mathlib's `Complex.digamma_one_half`.\n\nPart 2 (next): the digamma series from BohrMollerup's logGammaSeq and the\nj \u2265 1 polygamma values at 1/2, then the odd-m reindexing to lagariasA.\n\nArchive references: D-066 (route \u03b24, piece A), journal-019/020.\n-/\nimport Mathlib\nimport NguyenRH1.LiDigamma\n\nopen Zeta23 Complex Filter Topology\nopen scoped Nat\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. The Gamma\u211d logDeriv split -/\n\n/-- The cpow factor as an exponential: \u03c0^(\u2212w/2) = exp(\u2212(log \u03c0)/2\u00b7w). -/\nlemma pi_cpow_neg_half_eq_exp (w : \u2102) :\n (\u2191Real.pi : \u2102) ^ (-w / 2) = Complex.exp (-(Real.log Real.pi : \u2102) / 2 * w) := by\n rw [Complex.cpow_def_of_ne_zero (by exact_mod_cast Real.pi_ne_zero)]\n congr 1\n rw [\u2190 Complex.ofReal_log Real.pi_nonneg]\n ring_nf\n\n/-- The Gamma\u211d logDeriv split: for (1+s)/2 off the poles of \u0393,\n logDeriv Gamma\u211d (1+s) = \u2212(log \u03c0)/2 + (1/2)\u00b7\u03c8((1+s)/2). -/\nlemma logDeriv_Gamma\u211d_one_add {s : \u2102}\n (hs : Complex.Gamma ((1 + s) / 2) \u2260 0)\n (hs2 : \u2200 m : \u2115, ((1 + s) / 2) \u2260 -m) :\n logDeriv Complex.Gamma\u211d (1 + s) =\n -(Real.log Real.pi : \u2102) / 2 + (1 / 2) * Complex.digamma ((1 + s) / 2) := by\n have hfun : Complex.Gamma\u211d = fun w : \u2102 =>\n Complex.exp (-(Real.log Real.pi : \u2102) / 2 * w) * Complex.Gamma (w / 2) := by\n funext w\n show Complex.Gamma\u211d w = _\n rw [Complex.Gamma\u211d_def]\n congr 1\n exact pi_cpow_neg_half_eq_exp w\n rw [hfun]\n have hdif1 : DifferentiableAt \u2102 (fun w : \u2102 => Complex.exp (-(Real.log Real.pi : \u2102) / 2 * w))\n (1 + s) := by\n have h1 : DifferentiableAt \u2102 (fun w : \u2102 => -(Real.log Real.pi : \u2102) / 2 * w) (1 + s) :=\n differentiableAt_id.const_mul _\n exact Complex.differentiableAt_exp.comp (1 + s) h1\n have hdif2 : DifferentiableAt \u2102 (fun w : \u2102 => Complex.Gamma (w / 2)) (1 + s) := by\n have h2 : DifferentiableAt \u2102 (fun w : \u2102 => w / 2) (1 + s) :=\n differentiableAt_id.div_const 2\n exact (Complex.differentiableAt_Gamma _ hs2).comp (1 + s) h2\n rw [logDeriv_mul (f := fun w : \u2102 => Complex.exp (-(Real.log Real.pi : \u2102) / 2 * w))\n (g := fun w => Complex.Gamma (w / 2)) (1 + s) (Complex.exp_ne_zero _) hs hdif1 hdif2]\n have h1 : logDeriv (fun w : \u2102 => Complex.exp (-(Real.log Real.pi : \u2102) / 2 * w)) (1 + s) =\n -(Real.log Real.pi : \u2102) / 2 := by\n rw [logDeriv_apply]\n have hd : deriv (fun w : \u2102 => Complex.exp (-(Real.log Real.pi : \u2102) / 2 * w)) (1 + s) =\n Complex.exp (-(Real.log Real.pi : \u2102) / 2 * (1 + s)) * (-(Real.log Real.pi : \u2102) / 2) := by\n have hh1 : HasDerivAt (fun w : \u2102 => -(Real.log Real.pi : \u2102) / 2 * w)\n (-(Real.log Real.pi : \u2102) / 2) (1 + s) := by\n simpa using (hasDerivAt_id (1 + s)).const_mul (-(Real.log Real.pi : \u2102) / 2)\n have hh2 := (Complex.hasDerivAt_exp (-(Real.log Real.pi : \u2102) / 2 * (1 + s))).comp (1 + s) hh1\n have heq : (Complex.exp \u2218 fun w : \u2102 => -(Real.log Real.pi : \u2102) / 2 * w) =\n fun w : \u2102 => Complex.exp (-(Real.log Real.pi : \u2102) / 2 * w) := rfl\n rw [heq] at hh2\n exact hh2.deriv\n rw [hd]\n field_simp [Complex.exp_ne_zero (-(Real.log Real.pi : \u2102) / 2 * (1 + s))]\n have h2 : logDeriv (fun w : \u2102 => Complex.Gamma (w / 2)) (1 + s) =\n (1 / 2) * Complex.digamma ((1 + s) / 2) := by\n rw [logDeriv_apply]\n have hd : deriv (fun w : \u2102 => Complex.Gamma (w / 2)) (1 + s) =\n deriv Complex.Gamma ((1 + s) / 2) * (1 / 2) := by\n have hh1 : HasDerivAt (fun w : \u2102 => w / 2) (1 / 2) (1 + s) := by\n simpa using (hasDerivAt_id (1 + s)).div_const (2 : \u2102)\n have hh2 := ((Complex.differentiableAt_Gamma _ hs2).hasDerivAt).comp (1 + s) hh1\n have heq : (Complex.Gamma \u2218 fun w : \u2102 => w / 2) = fun w : \u2102 => Complex.Gamma (w / 2) := rfl\n rw [heq] at hh2\n exact hh2.deriv\n rw [hd, Complex.digamma_def, logDeriv_apply]\n field_simp [hs]\n rw [h1, h2]\n\n/-! ## 2. The j = 0 evaluation: \u03c4\u2080 = \u2212a\u2081 -/\n\n/-- log(4\u03c0) = 2\u00b7log 2 + log \u03c0. -/\nlemma log_four_pi : Real.log (4 * Real.pi) = 2 * Real.log 2 + Real.log Real.pi := by\n rw [Real.log_mul (by norm_num) (by positivity)]\n congr 1\n rw [show (4 : \u211d) = 2 ^ 2 from by norm_num, Real.log_pow]\n ring\n\n/-- **\u03c4\u2080** (D-066 \u00a74, j = 0): logDeriv Gamma\u211d 1 = \u2212(log 4\u03c0 + \u03b3)/2 = \u2212a\u2081. -/\ntheorem logDeriv_Gamma\u211d_one :\n logDeriv Complex.Gamma\u211d 1 = -(lagariasA1 : \u2102) := by\n have hs : Complex.Gamma ((1 + (0 : \u2102)) / 2) \u2260 0 := by\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e, Complex.Gamma_one_half_eq]\n have hpi : (\u2191Real.pi : \u2102) \u2260 0 := by exact_mod_cast Real.pi_ne_zero\n rw [Complex.cpow_def_of_ne_zero hpi]\n exact Complex.exp_ne_zero _\n have hs2 : \u2200 m : \u2115, ((1 + (0 : \u2102)) / 2) \u2260 -m := by\n intro m hm\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e] at hm\n have hre : ((1 : \u2102) / 2).re = (-(m : \u2102)).re := by rw [hm]\n simp at hre\n have hm0 : (0 : \u211d) \u2264 (m : \u211d) := Nat.cast_nonneg m\n linarith\n have h := logDeriv_Gamma\u211d_one_add (s := 0) hs hs2\n rw [add_zero] at h\n rw [h]\n rw [Complex.digamma_one_half]\n rw [show Complex.log (2 : \u2102) = (Real.log 2 : \u2102) by\n rw [show (2 : \u2102) = ((2 : \u211d) : \u2102) by simp]\n exact (Complex.ofReal_log (by norm_num : (0 : \u211d) \u2264 2)).symm]\n rw [lagariasA1, log_four_pi]\n push_cast\n ring\n\n/-! ## 3. The full archimedean coefficient evaluation (all j) -/\n\n/-- The split's hypotheses hold on a neighbourhood of 0. -/\nlemma eventually_Gamma_ne_zero_and_pole_free :\n (\u2200\u1da0 s in \ud835\udcdd (0 : \u2102), Complex.Gamma ((1 + s) / 2) \u2260 0) \u2227\n (\u2200\u1da0 s in \ud835\udcdd (0 : \u2102), \u2200 m : \u2115, (1 + s) / 2 \u2260 -(m : \u2102)) := by\n have hcont1 : ContinuousAt (fun s : \u2102 => Complex.Gamma ((1 + s) / 2)) 0 := by\n have haffine : ContinuousAt (fun s : \u2102 => (1 + s) / 2) 0 := by\n have h := (continuousAt_const.add continuousAt_id : ContinuousAt (fun s : \u2102 => 1 + s) 0)\n simpa using h.div_const (2 : \u2102)\n have hG : ContinuousAt Complex.Gamma ((1 + (0 : \u2102)) / 2) := by\n apply Complex.continuousAt_Gamma\n intro m\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e]\n intro hcontra\n have h2 : ((1 : \u2102) / 2).re = (-(m : \u2102)).re := by rw [hcontra]\n simp at h2\n have hm : (0 : \u211d) \u2264 (m : \u211d) := Nat.cast_nonneg m\n linarith\n have hcomp := hG.comp 0 haffine\n have heq : (Complex.Gamma \u2218 fun s : \u2102 => (1 + s) / 2) =\n fun s => Complex.Gamma ((1 + s) / 2) := rfl\n rw [heq] at hcomp\n exact hcomp\n have h1 : \u2200\u1da0 s in \ud835\udcdd (0 : \u2102), Complex.Gamma ((1 + s) / 2) \u2260 0 := by\n have hG1 : Complex.Gamma ((1 + (0 : \u2102)) / 2) \u2260 0 := by\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e, Complex.Gamma_one_half_eq]\n have hpi : (\u2191Real.pi : \u2102) \u2260 0 := by exact_mod_cast Real.pi_ne_zero\n rw [Complex.cpow_def_of_ne_zero hpi]\n exact Complex.exp_ne_zero _\n exact hcont1.tendsto.eventually_ne hG1\n have h2 : \u2200\u1da0 s in \ud835\udcdd (0 : \u2102), \u2200 m : \u2115, (1 + s) / 2 \u2260 -(m : \u2102) := by\n have hopen : IsOpen {s : \u2102 | 0 < (((1 + s) / 2).re)} := by\n have hcont : Continuous (fun s : \u2102 => (((1 + s) / 2).re)) := by\n have h1 : Continuous (fun s : \u2102 => (1 + s) / 2) :=\n (continuous_const.add continuous_id).div_const (2 : \u2102)\n exact Complex.continuous_re.comp h1\n exact isOpen_Ioi.preimage hcont\n have hmem : (0 : \u2102) \u2208 {s : \u2102 | 0 < (((1 + s) / 2).re)} := by\n show 0 < (((1 + (0 : \u2102)) / 2).re)\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e]\n simp\n filter_upwards [hopen.mem_nhds hmem] with s hs\n intro m hmeq\n rw [hmeq] at hs\n simp at hs\n have hm : (0 : \u211d) \u2264 (m : \u211d) := Nat.cast_nonneg m\n linarith\n exact \u27e8h1, h2\u27e9\n\n/-- **Piece (A), the archimedean Taylor coefficients (D-066 \u00a74)**: for all j,\n iteratedDeriv j (fun s => logDeriv Gamma\u211d (1+s)) 0 = (\u22121)^{j+1}\u00b7j!\u00b7\u2191lagariasA(j+1). -/\ntheorem iteratedDeriv_logDeriv_Gamma\u211d (j : \u2115) :\n iteratedDeriv j (fun s : \u2102 => logDeriv Complex.Gamma\u211d (1 + s)) 0 =\n ((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d) := by\n rcases Nat.eq_zero_or_pos j with hj0 | hj0\n \u00b7 subst hj0\n rw [iteratedDeriv_zero]\n rw [lagariasA, if_pos rfl]\n simp [logDeriv_Gamma\u211d_one]\n \u00b7 have hj : 1 \u2264 j := hj0\n obtain \u27e8hG, hpole\u27e9 := eventually_Gamma_ne_zero_and_pole_free\n have hev : (fun s : \u2102 => logDeriv Complex.Gamma\u211d (1 + s)) =\u1da0[\ud835\udcdd (0 : \u2102)]\n (fun s => -(Real.log Real.pi : \u2102) / 2 + (1 / 2) * Complex.digamma ((1 + s) / 2)) := by\n filter_upwards [hG, hpole] with s h1 h2\n exact logDeriv_Gamma\u211d_one_add h1 h2\n rw [iteratedDeriv_congr_of_eventuallyEq hev]\n -- the constant dies for j \u2265 1\n have hsplit2 : (fun s : \u2102 => -(Real.log Real.pi : \u2102) / 2 +\n (1 / 2) * Complex.digamma ((1 + s) / 2)) =\n (fun _ : \u2102 => -(Real.log Real.pi : \u2102) / 2) +\n (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) := rfl\n rw [hsplit2]\n have hcd0 : ContDiffAt \u2102 j (fun s => Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt \u2102 Complex.digamma ((1 : \u2102) / 2) :=\n digamma_analyticOn _ dpsiDomain_half\n have h2 : AnalyticAt \u2102 (fun s : \u2102 => (1 + s) / 2) 0 :=\n (analyticAt_const.add analyticAt_id).div analyticAt_const (by norm_num)\n have h1' : AnalyticAt \u2102 Complex.digamma ((fun s : \u2102 => (1 + s) / 2) 0) := by\n simpa using h1\n have h3 := h1'.comp h2\n have heq : (Complex.digamma \u2218 (fun s : \u2102 => (1 + s) / 2)) =\n fun s => Complex.digamma ((1 + s) / 2) := rfl\n rw [heq] at h3\n exact h3.contDiffAt\n have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=\n hcd0.const_mul _\n rw [iteratedDeriv_add contDiff_const.contDiffAt hcd]\n rw [iteratedDeriv_const, if_neg (by omega : j \u2260 0), zero_add]\n rw [iteratedDeriv_const_mul (1 / 2 : \u2102) hcd0]\n rw [iteratedDeriv_one_add_div_two Complex.digamma j (by\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e]\n exact digamma_analyticOn _ dpsiDomain_half) (by\n have hcont : Tendsto (fun s : \u2102 => (1 + s) / 2) (\ud835\udcdd 0) (\ud835\udcdd ((1 + (0 : \u2102)) / 2)) := by\n have hc : ContinuousAt (fun s : \u2102 => (1 + s) / 2) 0 := by\n have h := (continuousAt_const.add continuousAt_id : ContinuousAt (fun s : \u2102 => 1 + s) 0)\n simpa using h.div_const (2 : \u2102)\n exact hc.tendsto\n have h1 : ((1 + (0 : \u2102)) / 2) = 1 / 2 := by ring\n rw [h1] at hcont\n exact hcont.eventually ((digamma_analyticOn _ dpsiDomain_half).eventually_analyticAt))]\n rw [iteratedDeriv_digamma_half_value j hj]\n have h2pow : ((1 : \u2102) / 2) * ((1 : \u2102) / 2) ^ j * (2 : \u2102) ^ (j + 1) = 1 := by\n rw [show ((1 : \u2102) / 2) * ((1 : \u2102) / 2) ^ j = ((1 : \u2102) / 2) ^ (j + 1) from by\n rw [pow_succ]; ring]\n rw [\u2190 mul_pow]\n simp\n have hkey : (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j *\n (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * ((2 : \u2102) ^ (j + 1) * (lagariasA (j + 1) : \u211d)))) =\n ((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d) := by\n rw [show (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) *\n ((2 : \u2102) ^ (j + 1) * (lagariasA (j + 1) : \u211d)))) =\n (((1 : \u2102) / 2) * ((1 : \u2102) / 2) ^ j * (2 : \u2102) ^ (j + 1)) *\n (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d)) by ring]\n rw [h2pow]\n ring\n exact hkey\n",
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" digamma_analyticOn _ dpsiDomain_half",
" have h2 : AnalyticAt \u2102 (fun s : \u2102 => (1 + s) / 2) 0 :=",
" (analyticAt_const.add analyticAt_id).div analyticAt_const (by norm_num)",
"- have h1' : AnalyticAt \u2102 Complex.digamma ((fun s : \u2102 => (1 + s) / 2) 0) := by",
"- simpa using h1",
"- have h3 := h1'.comp h2",
"+ have h3 := AnalyticAt.comp (g := fun s : \u2102 => (1 + s) / 2) h1 h2",
" have heq : (Complex.digamma \u2218 (fun s : \u2102 => (1 + s) / 2)) =",
" fun s => Complex.digamma ((1 + s) / 2) := rfl",
" rw [heq] at h3",
" exact h3.contDiffAt",
" have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=",
"- hcd0.const_mul _",
"+ (contDiff_const (\ud835\udd5c := \u2102) (c := -(Real.log Real.pi : \u2102) / 2 * 0 + (1 / 2))).contDiffAt.mul hcd0",
" rw [iteratedDeriv_add contDiff_const.contDiffAt hcd]",
" rw [iteratedDeriv_const, if_neg (by omega : j \u2260 0), zero_add]",
" rw [iteratedDeriv_const_mul (1 / 2 : \u2102) hcd0]"
]
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[{"type":"thinking","thinking":"Issues:\n\n1. **Line 155**: `h.div_const (2 : ℂ)` — the error \"numerals are data... ContinuousAt (HDiv.hDiv (1+0)) 2 : Prop\" — the div_const's elaboration — `h.div_const (2:ℂ)` — ContinuousAt.div_const (hf : ContinuousAt f x) (c : 𝕜)? — hmm — the error suggests div_const isn't matching. Let me check ContinuousAt.div_const's signature — maybe it's `ContinuousAt.div_const (hf : ContinuousAt f x) (c : 𝕜) : ContinuousAt (fun y => f y / c) x`? — the error \"ContinuousAt (HDiv.hDiv (1+0)) 2\" — the elaborator parsed `h.div_const (2:ℂ)` as... hmm — maybe the method is different. Let me check what `.div_const` resolves to for ContinuousAt.\n\nActually — hold on, the error at 155 is in hcont1's haffine: `simpa using h.div_const (2 : ℂ)` — and the error shows `ContinuousAt (HDiv.hDiv (1 + 0)) 2 : Prop` — the div_const application elaborated h.div_const's argument... — let me check if ContinuousAt.div_const exists. If not, use: `(continuousAt_const.add continuousAt_id).div continuousAt_const two_ne_zero`? — the ContinuousAt.div needs the side — `ContinuousAt.div (hf) (hg) (h : g x ≠ 0)`? — the side (fun _ => 2) 0 ≠ 0 — `by norm_num`... — earlier that errored differently. Let me check the names quickly.\n\n2. **Line 218**: h1'.comp h2 — the h1' : AnalyticAt ℂ digamma ((fun s => (1+s)/2) 0) — and h2 : AnalyticAt ℂ (fun s => (1+s)/2) 0 — the comp: AnalyticAt.comp (hf : AnalyticAt f (g x)) (hg : AnalyticAt g x) — with g := fun s => (1+s)/2, x := 0: g x = (fun s => (1+s)/2) 0 — and h1's point must be exactly that application form `(fun s => (1+s)/2) 0` — my h1' HAS that form ✓ — but the error says h2 \"is expected to have type AnalyticAt ℂ (HDiv.hDiv (1+0)) 2\" — so the elaborator unified g := HDiv.hDiv (1+0)?!?! — because h1''s point `(fun s => (1+s)/2) 0` — Lean read the APPLICATION form and tried to decompose it as (g 0) with g := HDiv.hDiv (1+0) (treating the application as (something) 0!) — since `(fun s => (1+s)/2) 0` is an application of (fun s => ...) to 0 — Lean's unifier for h1'.comp h2 needs (g ?x) with ?x — the point in h1' is the raw application (fun s => (1+s)/2) 0 — the unifier might beta-reduce it to (1+0)/2 then fail to match against (g 0) with g := fun s => (1+s)/2 — should be FINE (beta)... — the error suggests the unification assigned g := HDiv.hDiv (1+0) — i.e., it parsed h1''s point as `HDiv.hDiv (1+0) 2`?? — no wait — the expected type for h2 is `AnalyticAt ℂ (HDiv.hDiv (1+0)) 2` — that's AnalyticAt of the function `HDiv.hDiv (1+0)` at point `2`!! — the unifier saw h1''s point as (fun s => (1+s)/2) 0 and decomposed... no. It decomposed h1''s point `((fun s => (1+s)/2)) 0` — hmm — the expected type of h2 is AnalyticAt ℂ (HDiv.hDiv (1+0)) 2 — so the unifier set g := (HDiv.hDiv (1+0))?? and x := 2 — i.e., it read h1''s point as `(HDiv.hDiv (1+0)) 2`?? — that means h1''s point elaborated to `HDiv.hDiv (1+0) 2` — WAIT — h1''s point I wrote as `((fun s : ℂ => (1 + s) / 2) 0)` — that's ((fun s => (1+s)/2)) applied to 0 — beta — (1+0)/2 — hmm — but the ERROR says the expected h2 type has point 2 and function HDiv.hDiv (1+0) — so Lean parsed h1''s point `((fun s : ℂ => (1 + s) / 2) 0)` as `HDiv.hDiv (1+0) 2`?? — no — `(fun s => (1+s)/2) 0` — how would that become HDiv.hDiv (1+0) 2... — `(1+s)/2` with s := 0: (1+0)/2 = HDiv.hDiv (1+0) 2 — YES — that's the beta-REDUCED form — and h1' has that as its point — so h1' : AnalyticAt ℂ digamma (HDiv.hDiv (1+0) 2) — i.e. digamma ((1+0)/2) — and the comp's unification: AnalyticAt.comp wants hf : AnalyticAt f (g x) — with h1''s point being (1+0)/2 = HDiv.hDiv (1+0) 2 — the unifier needs to see it as (g x) — it chose g := HDiv.hDiv (1+0), x := 2?!?! — because `(1+0)/2` as `(g x)` with g := `HDiv.hDiv (1+0)` and x := 2 — YES that's a valid first-order match (the term (1+0)/2 IS HDiv.hDiv (1+0) 2 — with the last-arg-as-application pattern)!! The unifier prefers that wrong match. To force the right one: state h1''s point as literally `(fun s : ℂ => (1+s)/2) 0` UNREDUCED — I did that with the simpa — but simpa normalized it to (1+0)/2 = HDiv.hDiv (1+0) 2 — the beta-reduced form — which then mis-unifies. Fix: DON'T let it normalize — use `show`:\n```lean\n have h1' : AnalyticAt ℂ Complex.digamma ((fun s : ℂ => (1 + s) / 2) 0) := by\n show AnalyticAt ℂ Complex.digamma ((1 + (0:ℂ))/2) -- beta of the application — hmm the point is the application (fun s => (1+s)/2) 0 — and I show (1+0)/2 — these are defeq — the SHOW changes h1''s type to the (1+0)/2 form — same problem.\n```\nThe problem: I NEED h1''s point to be the UNREDUCED application `(fun s => (1+s)/2) 0` for the comp's unifier to pick g := fun s => (1+s)/2. But any tactic normalizes it. Alternative: avoid the issue by using `AnalyticAt.comp'`? or by specifying g via the comp's implicit args: `AnalyticAt.comp (f := Complex.digamma) (g := fun s : ℂ => (1+s)/2) h1 h2`? — if the f/g are implicit named args I can pin them:\n```lean\n exact AnalyticAt.comp (g := fun s : ℂ => (1 + s) / 2) h1 h2 |>.contDiffAt\n```\nhmm — AnalyticAt.comp's explicit args are (hf) (hg) — the f, g, x implicit — with g pinned, the unification of h1's point (1/2 — via my h1's direct form) with (g 0) = (fun s => (1+s)/2) 0 — beta to (1+0)/2 — vs h1's point 1/2 — defeq ✓ — the explicit g-pinning should force the right match. Let me use:\n```lean\n exact (AnalyticAt.comp (g := fun s : ℂ => (1 + s) / 2) h1 h2).contDiffAt\n```\nwith h1 : AnalyticAt ℂ Complex.digamma (1/2) directly (no h1' detour) — the (g 0) = (fun s => (1+s)/2) 0 — beta → (1+0)/2 — vs h1's 1/2 — defeq ✓ — hmm — will the elaborator reduce (fun s => (1+s)/2) 0 to (1+0)/2 then to 1/2? — for the defeq check it reduces ✓ should pass. And the resulting type: AnalyticAt ℂ (digamma ∘ (fun s => (1+s)/2)) 0 — then the heq bridge to the lambda ✓ then .contDiffAt ✓.\n\n3. **Line 224**: `hcd0.const_mul _` — ContDiffAt.const_mul — \"Function.const_mul does not have a usable parameter\" — wrong namespace resolution — the method: `ContDiffAt.const_mul (c) (h : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (fun y => c * f y) x` — dot-form h.const_mul c? — or the const_mul has args (c) then the ContDiffAt — `hcd0.const_mul` — hcd0 : ContDiffAt — hcd0.const_mul — should resolve ContDiffAt.const_mul ✓ — the error says Function.const_mul — the elaborator found Function.const_mul (a different thing!) — because ContDiffAt.const_mul doesn't exist with that name? — the iteratedDeriv_const_mul exists (used in LiResidue) — for ContDiffAt: `ContDiffAt.const_mul`? — hmm — maybe it's `ContDiffAt.const_mul (c : 𝔸) (hf : ContDiffAt 𝕜 n f x)` — the arg order (c first) — so `hcd0.const_mul _` would still work via dot... the error shows Function.const_mul — meaning ContDiffAt.const_mul wasn't found and it fell to Function.const_mul?? no — dot-notation for hcd0 : ContDiffAt — resolves ContDiffAt.const_mul first — if missing, error \"unknown constant ContDiffAt.const_mul\" — the error instead tried Function.const_mul — bizarre. Maybe the name is `ContDiffAt.const_mul` — let me check. Or use `ContDiffAt.mul (contDiff_const.contDiffAt) hcd0`? — that gives (const * f) — the Pi.mul form — my target (fun s => (1/2) * digamma(...)) — the lambda-of-product — ContDiffAt.mul : ContDiffAt (f * g) — Pi.mul — vs the lambda — defeq via Pi.mul_apply ✓. Use:\n```lean\n have hcd : ContDiffAt ℂ j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=\n (contDiff_const (𝕜 := ℂ)).contDiffAt.mul hcd0\n```\nhmm — contDiff_const : ContDiff 𝕜 n fun _ => c — .contDiffAt : ContDiffAt — then .mul hcd0 : ContDiffAt ((fun _ => c) * (digamma-comp)) 0 — Pi.mul — the target lambda — defeq ✓. But the c's inference — `contDiff_const` needs the c — from the target's ascription ✓ (hcd's stated type pins it) ✓.\n\nLet me apply these.","signature":""},{"type":"text","text":"Fixing the comp unification (pin g) and the const_mul resolution."},{"type":"tool_use","id":"Edit_564","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean","old_string":" have hcd0 : ContDiffAt ℂ j (fun s => Complex.digamma ((1 + s) / 2)) 0 := by\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 h1' : AnalyticAt ℂ Complex.digamma ((fun s : ℂ => (1 + s) / 2) 0) := by\n simpa using h1\n have h3 := h1'.comp h2\n have heq : (Complex.digamma ∘ (fun s : ℂ => (1 + s) / 2)) =\n fun s => Complex.digamma ((1 + s) / 2) := rfl\n rw [heq] at h3\n exact h3.contDiffAt\n have hcd : ContDiffAt ℂ j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=\n hcd0.const_mul _","new_string":" have hcd0 : ContDiffAt ℂ j (fun s => Complex.digamma ((1 + s) / 2)) 0 := by\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 (g := fun s : ℂ => (1 + s) / 2) h1 h2\n have heq : (Complex.digamma ∘ (fun s : ℂ => (1 + s) / 2)) =\n fun s => Complex.digamma ((1 + s) / 2) := rfl\n rw [heq] at h3\n exact h3.contDiffAt\n have hcd : ContDiffAt ℂ j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 :=\n (contDiff_const (𝕜 := ℂ) (c := -(Real.log Real.pi : ℂ) / 2 * 0 + (1 / 2))).contDiffAt.mul hcd0"},"extra_content":null}]