← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (4422 chars)
The hrw is backwards — I need: the goal after all rewrites is:
`(1/2) * ((1/2)^j * ((−1)^{j+1}·j!·(2^{j+1}·↑lagariasA)))` = the target `(−1)^{j+1}·j!·↑lagariasA` — wait — let me trace the goal at the end:
After `rw [iteratedDeriv_congr_of_eventuallyEq hev]` — the goal: iteratedDeriv j (split-form) 0 = (−1)^{j+1}·j!·↑lagariasA(j+1).
After hsplit2 + iteratedDeriv_add + iteratedDeriv_const if_neg + zero_add: goal: iteratedDeriv j (fun s => (1/2)·digamma((1+s)/2)) 0 = (−1)^{j+1}·j!·↑lagariasA(j+1).
After iteratedDeriv_const_mul (1/2) hcd0: (1/2)·iteratedDeriv j (digamma∘φ) 0 = target.
After iteratedDeriv_one_add_div_two: (1/2)·((1/2)^j·iteratedDeriv j digamma (1/2)) = target.
After iteratedDeriv_digamma_half_value: (1/2)·((1/2)^j·((−1)^{j+1}·j!·(2^{j+1}·↑lagariasA(j+1)))) = (−1)^{j+1}·j!·↑lagariasA(j+1).
So the FINAL goal is:
(1/2) * ((1/2)^j * ((−1)^{j+1}·↑j!·(2^{j+1}·↑lagariasA(j+1)))) = (−1)^{j+1}·↑j!·↑lagariasA(j+1)
The 2-cancellation: (1/2)·(1/2)^j·2^{j+1} = (1/2)^{j+1}·2^{j+1} = (2/2)^{j+1} = 1 ✓. So prove the goal via:
```lean
have h2pow : ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j * (2 : ℂ) ^ (j + 1) = 1 := by
rw [show ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j = ((1 : ℂ) / 2) ^ (j + 1) from by ring]
rw [← mul_pow]
norm_num -- (2·(1/2))^{j+1} = 1^{j+1} = 1
have hkey : (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j *
(((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =
((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ) := by
rw [show (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j * (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) *
((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =
((1 / 2) * ((1 / 2) ^ j * (2 : ℂ) ^ (j + 1))) *
(((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ)) by ring]
rw [h2pow]
ring
exact hkey
```
Wait — the goal after the last rw is EXACTLY hkey's LHS = hkey's RHS — so `exact hkey` — but my code has `rw [hrw]` with hrw backwards — replace with exact hkey. Let me write:
```lean
rw [iteratedDeriv_digamma_half_value j hj]
have h2pow : ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j * (2 : ℂ) ^ (j + 1) = 1 := by
rw [show ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j = ((1 : ℂ) / 2) ^ (j + 1) from by ring]
rw [← mul_pow]
norm_num
have hkey : (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j *
(((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =
((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ) := by
rw [show (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j * (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) *
((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =
((1 / 2 : ℂ) * ((1 : ℂ) / 2) ^ j * (2 : ℂ) ^ (j + 1)) *
(((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ)) by ring]
rw [h2pow]
ring
exact hkey
```
Hmm — the h2pow's norm_num: ((1/2)·2)^(j+1) = 1: (1/2·2) = 1: `norm_num` — (1/2·2)^{j+1} — norm_num should simplify the base 1/2·2 = 1 then 1^{j+1} = 1 ✓ — `norm_num` with pow — the exponent j+1 is a variable — norm_num simplifies the BASE then one_pow ✓ — maybe need `simp` instead: (1/2·2:ℂ)^(j+1) = 1^(j+1) [norm_num for the base] = 1 [one_pow] — `simp` handles both ✓. Use `simp`.
Also the h2pow's first show: (1/2)·(1/2)^j = (1/2)^{j+1} — by ring? — x·x^j = x^{j+1} — ring handles? — ring knows x·x^j = x^{j+1}? — for a semiring with variable exponent — ring treats x^j as atom: x·(x^j) vs x^{j+1} — ring does NOT normalize pow-of-variable-exponent... — x^{j+1} = x^j·x via pow_succ — ring's normal form for x^{j+1}... — hmm — for variable exponents, ring might not. Use `pow_succ` reversed: x^{j+1} = x^j·x [pow_succ] so the show: (1/2)·(1/2)^j = (1/2)^{j+1} ⟸ rw [pow_succ]; ring ✓:
```lean
rw [show ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j = ((1 : ℂ) / 2) ^ (j + 1) from by
rw [pow_succ]; ring]
rw [← mul_pow]
simp
```
pow_succ : a^(n+1) = a^n·a — the show's RHS (1/2)^{j+1} → (1/2)^j·(1/2) ✓ then ring: (1/2)·(1/2)^j = (1/2)^j·(1/2) ✓ mul_comm ✓.
And hkey's big show — the reassociation (1/2)·((1/2)^j·(S·(2^{j+1}·a))) = ((1/2)·(1/2)^j·2^{j+1})·(S·a) with S := (−1)^{j+1}·j! — ring ✓ (all atoms).
Then after rw [h2pow]: (1)·(S·a) = S·a — wait — hkey's proof: the show-rewrite then rw [h2pow] — h2pow's LHS pattern (1/2)·(1/2)^j·2^{j+1} — matches the rewritten form ✓ → 1·(S·a)... hmm — then `ring`: 1·(S·a) = S·a ✓ closes.
Let me now rewrite the whole tail block from 211.
Rewriting the whole tail block of the capstone with all fixes.
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean",
"old_string": " have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt \u2102 Complex.digamma ((1 + (0 : \u2102)) / 2) := by\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e]\n exact 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\n exact (h1.comp h2).contDiffAt.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 _ hcd]\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 :=\n (continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num)\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 -- the 2-power cancellation: (1/2)\u00b7(1/2)^j\u00b72^{j+1} = 1\n have h2pow : (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (2 : \u2102) ^ (j + 1)) = 1 := by\n rw [show (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (2 : \u2102) ^ (j + 1)) =\n ((1 / 2) * 2) ^ (j + 1) by rw [mul_pow]; ring_nf]\n norm_num\n have hrw : (-1 : \u2102) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d) =\n (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j *\n (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * ((2 : \u2102) ^ (j + 1) * (lagariasA (j + 1) : \u211d)))) := by\n rw [h2pow] -- hmm \u2014 the 2-powers don't align this way\n ring\n rw [hrw]",
"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 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"
}
result 2026-08-22 11:07:26Z → 2026-08-22 11:07:26Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean",
"oldString": " have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt \u2102 Complex.digamma ((1 + (0 : \u2102)) / 2) := by\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e]\n exact 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\n exact (h1.comp h2).contDiffAt.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 _ hcd]\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 :=\n (continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num)\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 -- the 2-power cancellation: (1/2)\u00b7(1/2)^j\u00b72^{j+1} = 1\n have h2pow : (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (2 : \u2102) ^ (j + 1)) = 1 := by\n rw [show (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (2 : \u2102) ^ (j + 1)) =\n ((1 / 2) * 2) ^ (j + 1) by rw [mul_pow]; ring_nf]\n norm_num\n have hrw : (-1 : \u2102) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d) =\n (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j *\n (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * ((2 : \u2102) ^ (j + 1) * (lagariasA (j + 1) : \u211d)))) := by\n rw [h2pow] -- hmm \u2014 the 2-powers don't align this way\n ring\n rw [hrw]",
"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 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",
"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 hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt \u2102 Complex.digamma ((1 + (0 : \u2102)) / 2) := by\n have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring\n rw [e]\n exact 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\n exact (h1.comp h2).contDiffAt.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 _ hcd]\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 :=\n (continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num)\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 -- the 2-power cancellation: (1/2)\u00b7(1/2)^j\u00b72^{j+1} = 1\n have h2pow : (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (2 : \u2102) ^ (j + 1)) = 1 := by\n rw [show (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (2 : \u2102) ^ (j + 1)) =\n ((1 / 2) * 2) ^ (j + 1) by rw [mul_pow]; ring_nf]\n norm_num\n have hrw : (-1 : \u2102) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d) =\n (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j *\n (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * ((2 : \u2102) ^ (j + 1) * (lagariasA (j + 1) : \u211d)))) := by\n rw [h2pow] -- hmm \u2014 the 2-powers don't align this way\n ring\n rw [hrw]\n",
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"lines": [
" (fun _ : \u2102 => -(Real.log Real.pi : \u2102) / 2) +",
" (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) := rfl",
" rw [hsplit2]",
"- have hcd : ContDiffAt \u2102 j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 := by",
"- have h1 : AnalyticAt \u2102 Complex.digamma ((1 + (0 : \u2102)) / 2) := by",
"- have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring",
"- rw [e]",
"- exact digamma_analyticOn _ dpsiDomain_half",
"+ have hcd0 : ContDiffAt \u2102 j (fun s => Complex.digamma ((1 + s) / 2)) 0 := by",
"+ have h1 : AnalyticAt \u2102 Complex.digamma ((1 : \u2102) / 2) :=",
"+ digamma_analyticOn _ dpsiDomain_half",
" have h2 : AnalyticAt \u2102 (fun s : \u2102 => (1 + s) / 2) 0 :=",
"- (analyticAt_const.add analyticAt_id).div analyticAt_const",
"- exact (h1.comp h2).contDiffAt.const_mul _",
"+ (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 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 _",
" rw [iteratedDeriv_add contDiff_const.contDiffAt hcd]",
" rw [iteratedDeriv_const, if_neg (by omega : j \u2260 0), zero_add]",
"- rw [iteratedDeriv_const_mul _ hcd]",
"+ rw [iteratedDeriv_const_mul (1 / 2 : \u2102) hcd0]",
" rw [iteratedDeriv_one_add_div_two Complex.digamma j (by",
" have e : (1 + (0 : \u2102)) / 2 = 1 / 2 := by ring",
" rw [e]",
" exact digamma_analyticOn _ dpsiDomain_half) (by",
" have hcont : Tendsto (fun s : \u2102 => (1 + s) / 2) (\ud835\udcdd 0) (\ud835\udcdd ((1 + (0 : \u2102)) / 2)) := by",
"- have hc : ContinuousAt (fun s : \u2102 => (1 + s) / 2) 0 :=",
"- (continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num)",
"+ have hc : ContinuousAt (fun s : \u2102 => (1 + s) / 2) 0 := by",
"+ have h := (continuousAt_const.add continuousAt_id : ContinuousAt (fun s : \u2102 => 1 + s) 0)",
"+ simpa using h.div_const (2 : \u2102)",
" exact hc.tendsto",
" have h1 : ((1 + (0 : \u2102)) / 2) = 1 / 2 := by ring",
" rw [h1] at hcont",
" exact hcont.eventually ((digamma_analyticOn _ dpsiDomain_half).eventually_analyticAt))]",
" rw [iteratedDeriv_digamma_half_value j hj]",
"- -- the 2-power cancellation: (1/2)\u00b7(1/2)^j\u00b72^{j+1} = 1",
"- have h2pow : (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (2 : \u2102) ^ (j + 1)) = 1 := by",
"- rw [show (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (2 : \u2102) ^ (j + 1)) =",
"- ((1 / 2) * 2) ^ (j + 1) by rw [mul_pow]; ring_nf]",
"- norm_num",
"- have hrw : (-1 : \u2102) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d) =",
"- (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j *",
"- (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * ((2 : \u2102) ^ (j + 1) * (lagariasA (j + 1) : \u211d)))) := by",
"- rw [h2pow] -- hmm \u2014 the 2-powers don't align this way",
"+ have h2pow : ((1 : \u2102) / 2) * ((1 : \u2102) / 2) ^ j * (2 : \u2102) ^ (j + 1) = 1 := by",
"+ rw [show ((1 : \u2102) / 2) * ((1 : \u2102) / 2) ^ j = ((1 : \u2102) / 2) ^ (j + 1) from by",
"+ rw [pow_succ]; ring]",
"+ rw [\u2190 mul_pow]",
"+ simp",
"+ have hkey : (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j *",
"+ (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * ((2 : \u2102) ^ (j + 1) * (lagariasA (j + 1) : \u211d)))) =",
"+ ((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d) := by",
"+ rw [show (1 / 2 : \u2102) * ((1 / 2 : \u2102) ^ j * (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) *",
"+ ((2 : \u2102) ^ (j + 1) * (lagariasA (j + 1) : \u211d)))) =",
"+ (((1 : \u2102) / 2) * ((1 : \u2102) / 2) ^ j * (2 : \u2102) ^ (j + 1)) *",
"+ (((-1 : \u2102)) ^ (j + 1) * (j ! : \u2102) * (lagariasA (j + 1) : \u211d)) by ring]",
"+ rw [h2pow]",
" ring",
"- rw [hrw]",
"+ exact hkey",
" "
]
}
],
"userModified": false,
"replaceAll": false
}
raw content_json (database row)
[{"type":"thinking","thinking":"The hrw is backwards — I need: the goal after all rewrites is:\n`(1/2) * ((1/2)^j * ((−1)^{j+1}·j!·(2^{j+1}·↑lagariasA)))` = the target `(−1)^{j+1}·j!·↑lagariasA` — wait — let me trace the goal at the end:\n\nAfter `rw [iteratedDeriv_congr_of_eventuallyEq hev]` — the goal: iteratedDeriv j (split-form) 0 = (−1)^{j+1}·j!·↑lagariasA(j+1).\nAfter hsplit2 + iteratedDeriv_add + iteratedDeriv_const if_neg + zero_add: goal: iteratedDeriv j (fun s => (1/2)·digamma((1+s)/2)) 0 = (−1)^{j+1}·j!·↑lagariasA(j+1).\nAfter iteratedDeriv_const_mul (1/2) hcd0: (1/2)·iteratedDeriv j (digamma∘φ) 0 = target.\nAfter iteratedDeriv_one_add_div_two: (1/2)·((1/2)^j·iteratedDeriv j digamma (1/2)) = target.\nAfter iteratedDeriv_digamma_half_value: (1/2)·((1/2)^j·((−1)^{j+1}·j!·(2^{j+1}·↑lagariasA(j+1)))) = (−1)^{j+1}·j!·↑lagariasA(j+1).\nSo the FINAL goal is:\n (1/2) * ((1/2)^j * ((−1)^{j+1}·↑j!·(2^{j+1}·↑lagariasA(j+1)))) = (−1)^{j+1}·↑j!·↑lagariasA(j+1)\nThe 2-cancellation: (1/2)·(1/2)^j·2^{j+1} = (1/2)^{j+1}·2^{j+1} = (2/2)^{j+1} = 1 ✓. So prove the goal via:\n```lean\n have h2pow : ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j * (2 : ℂ) ^ (j + 1) = 1 := by\n rw [show ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j = ((1 : ℂ) / 2) ^ (j + 1) from by ring]\n rw [← mul_pow]\n norm_num -- (2·(1/2))^{j+1} = 1^{j+1} = 1\n have hkey : (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j *\n (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =\n ((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ) := by\n rw [show (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j * (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) *\n ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =\n ((1 / 2) * ((1 / 2) ^ j * (2 : ℂ) ^ (j + 1))) *\n (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ)) by ring]\n rw [h2pow]\n ring\n exact hkey\n```\nWait — the goal after the last rw is EXACTLY hkey's LHS = hkey's RHS — so `exact hkey` — but my code has `rw [hrw]` with hrw backwards — replace with exact hkey. Let me write:\n```lean\n rw [iteratedDeriv_digamma_half_value j hj]\n have h2pow : ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j * (2 : ℂ) ^ (j + 1) = 1 := by\n rw [show ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j = ((1 : ℂ) / 2) ^ (j + 1) from by ring]\n rw [← mul_pow]\n norm_num\n have hkey : (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j *\n (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =\n ((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ) := by\n rw [show (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j * (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) *\n ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =\n ((1 / 2 : ℂ) * ((1 : ℂ) / 2) ^ j * (2 : ℂ) ^ (j + 1)) *\n (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ)) by ring]\n rw [h2pow]\n ring\n exact hkey\n```\nHmm — the h2pow's norm_num: ((1/2)·2)^(j+1) = 1: (1/2·2) = 1: `norm_num` — (1/2·2)^{j+1} — norm_num should simplify the base 1/2·2 = 1 then 1^{j+1} = 1 ✓ — `norm_num` with pow — the exponent j+1 is a variable — norm_num simplifies the BASE then one_pow ✓ — maybe need `simp` instead: (1/2·2:ℂ)^(j+1) = 1^(j+1) [norm_num for the base] = 1 [one_pow] — `simp` handles both ✓. Use `simp`.\n\nAlso the h2pow's first show: (1/2)·(1/2)^j = (1/2)^{j+1} — by ring? — x·x^j = x^{j+1} — ring handles? — ring knows x·x^j = x^{j+1}? — for a semiring with variable exponent — ring treats x^j as atom: x·(x^j) vs x^{j+1} — ring does NOT normalize pow-of-variable-exponent... — x^{j+1} = x^j·x via pow_succ — ring's normal form for x^{j+1}... — hmm — for variable exponents, ring might not. Use `pow_succ` reversed: x^{j+1} = x^j·x [pow_succ] so the show: (1/2)·(1/2)^j = (1/2)^{j+1} ⟸ rw [pow_succ]; ring ✓:\n```lean\n rw [show ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j = ((1 : ℂ) / 2) ^ (j + 1) from by\n rw [pow_succ]; ring]\n rw [← mul_pow]\n simp\n```\npow_succ : a^(n+1) = a^n·a — the show's RHS (1/2)^{j+1} → (1/2)^j·(1/2) ✓ then ring: (1/2)·(1/2)^j = (1/2)^j·(1/2) ✓ mul_comm ✓.\n\nAnd hkey's big show — the reassociation (1/2)·((1/2)^j·(S·(2^{j+1}·a))) = ((1/2)·(1/2)^j·2^{j+1})·(S·a) with S := (−1)^{j+1}·j! — ring ✓ (all atoms).\n\nThen after rw [h2pow]: (1)·(S·a) = S·a — wait — hkey's proof: the show-rewrite then rw [h2pow] — h2pow's LHS pattern (1/2)·(1/2)^j·2^{j+1} — matches the rewritten form ✓ → 1·(S·a)... hmm — then `ring`: 1·(S·a) = S·a ✓ closes.\n\nLet me now rewrite the whole tail block from 211.","signature":""},{"type":"text","text":"Rewriting the whole tail block of the capstone with all fixes."},{"type":"tool_use","id":"Edit_562","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean","old_string":" have hcd : ContDiffAt ℂ j (fun s => (1 / 2) * Complex.digamma ((1 + s) / 2)) 0 := by\n have h1 : AnalyticAt ℂ Complex.digamma ((1 + (0 : ℂ)) / 2) := by\n have e : (1 + (0 : ℂ)) / 2 = 1 / 2 := by ring\n rw [e]\n exact 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\n exact (h1.comp h2).contDiffAt.const_mul _\n rw [iteratedDeriv_add contDiff_const.contDiffAt hcd]\n rw [iteratedDeriv_const, if_neg (by omega : j ≠ 0), zero_add]\n rw [iteratedDeriv_const_mul _ hcd]\n rw [iteratedDeriv_one_add_div_two Complex.digamma j (by\n have e : (1 + (0 : ℂ)) / 2 = 1 / 2 := by ring\n rw [e]\n exact digamma_analyticOn _ dpsiDomain_half) (by\n have hcont : Tendsto (fun s : ℂ => (1 + s) / 2) (𝓝 0) (𝓝 ((1 + (0 : ℂ)) / 2)) := by\n have hc : ContinuousAt (fun s : ℂ => (1 + s) / 2) 0 :=\n (continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num)\n exact hc.tendsto\n have h1 : ((1 + (0 : ℂ)) / 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 -- the 2-power cancellation: (1/2)·(1/2)^j·2^{j+1} = 1\n have h2pow : (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j * (2 : ℂ) ^ (j + 1)) = 1 := by\n rw [show (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j * (2 : ℂ) ^ (j + 1)) =\n ((1 / 2) * 2) ^ (j + 1) by rw [mul_pow]; ring_nf]\n norm_num\n have hrw : (-1 : ℂ) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ) =\n (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j *\n (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) := by\n rw [h2pow] -- hmm — the 2-powers don't align this way\n ring\n rw [hrw]","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 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 _\n rw [iteratedDeriv_add contDiff_const.contDiffAt hcd]\n rw [iteratedDeriv_const, if_neg (by omega : j ≠ 0), zero_add]\n rw [iteratedDeriv_const_mul (1 / 2 : ℂ) hcd0]\n rw [iteratedDeriv_one_add_div_two Complex.digamma j (by\n have e : (1 + (0 : ℂ)) / 2 = 1 / 2 := by ring\n rw [e]\n exact digamma_analyticOn _ dpsiDomain_half) (by\n have hcont : Tendsto (fun s : ℂ => (1 + s) / 2) (𝓝 0) (𝓝 ((1 + (0 : ℂ)) / 2)) := by\n have hc : ContinuousAt (fun s : ℂ => (1 + s) / 2) 0 := by\n have h := (continuousAt_const.add continuousAt_id : ContinuousAt (fun s : ℂ => 1 + s) 0)\n simpa using h.div_const (2 : ℂ)\n exact hc.tendsto\n have h1 : ((1 + (0 : ℂ)) / 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 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j * (2 : ℂ) ^ (j + 1) = 1 := by\n rw [show ((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j = ((1 : ℂ) / 2) ^ (j + 1) from by\n rw [pow_succ]; ring]\n rw [← mul_pow]\n simp\n have hkey : (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j *\n (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =\n ((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ) := by\n rw [show (1 / 2 : ℂ) * ((1 / 2 : ℂ) ^ j * (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) *\n ((2 : ℂ) ^ (j + 1) * (lagariasA (j + 1) : ℝ)))) =\n (((1 : ℂ) / 2) * ((1 : ℂ) / 2) ^ j * (2 : ℂ) ^ (j + 1)) *\n (((-1 : ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ)) by ring]\n rw [h2pow]\n ring\n exact hkey"},"extra_content":null}]