← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (3159 chars)
The code at 768 looks right. The issue must be the `by omega` in the rw argument: `iteratedDerivWithin_dpsiTermC j (by omega) n hr` — when used inside `rw [...]`, the `by omega` proof obligation: 1 ≤ j — with hj0 : j > 0 in context ✓ — hmm — actually, wait: the rcases at 757: `rcases Nat.eq_zero_or_pos j with hj0 | hj0` — in the second branch hj0 : 0 < j — and `(by omega : 1 ≤ j)` — omega with hj0 : 0 < j — 1 ≤ j ⟺ 0 < j ✓ should work...
Hmm — actually maybe the issue is different: the rw's term is elaborated with the `by omega` needing the goal... — when you write `rw [lemma (by omega) ...]`, the by-block is elaborated with the expected type 1 ≤ j — that's fine.
Let me look at the error once more: "Did not find an occurrence of the pattern iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r in the target expression DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r" — the two ARE the same up to binder name... — UNLESS the pattern in the error display is what's generated from the lemma — and the target's — hmm — actually — Lean 4's rw DOES match up to alpha? Let me not assume — the classic behavior: rw matching is keyed by `kabstract` — which by default matches up to reducible defeq — and alpha-equivalence is NOT defeq (de Bruijn terms are equal but kabstract compares... — actually kabstract DOES handle alpha since de Bruijn makes binder names irrelevant — the displayed binder names are just pretty-printing). So alpha is not the issue.
WAIT — I bet it's this: the lemma's conclusion LHS: `iteratedDerivWithin k (fun s : ℂ => dpsiTermC s m) dpsiDomain s` — but the GOAL's term: `iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain r` — with the DIFFERENCE: my lemma's function is `(fun s : ℂ => dpsiTermC s m)` — hmm same...
Hmm hold on — maybe the error is that `rw [iteratedDerivWithin_dpsiTermC j (by omega) n hr]` — the `by omega` inside a rw argument creates a NEW GOAL (the side condition isn't discharged inline)?? — no — `by omega` fully elaborates inline when the expected type is known (1 ≤ j).
OK let me look at the actual error text once more, super carefully: "Did not find an occurrence of the pattern `iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r` in the target expression `DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r`" — hmm — IDENTICAL modulo binder. UNLESS — the `dpsiDomain` in the goal vs the lemma — dpsiDomain is a def — both reference the same def ✓. Or `iteratedDerivWithin` — same.
Let me try a completely different approach: convert the goal first via `show`:
```lean
· have h4 : iteratedDerivWithin j (fun z : ℂ => dpsiTermC z n) dpsiDomain r =
-((-1 : ℂ)) ^ j * (j ! : ℂ) * (r + ↑n + 1) ^ (-(j : ℤ) - 1) :=
iteratedDerivWithin_dpsiTermC j (by omega) n hr
rw [h4]
...
```
If the application itself elaborates fine, the rw [h4] uses h4's LHS as the pattern — h4's LHS is written exactly as the goal's term ✓ (my text) — then it matches by construction. The earlier failure might be from how the lemma-application's elaboration normalized something. Let me use the explicit h4.
The lemma application in rw is misbehaving — using an explicitly-typed `have` instead.
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"old_string": " \u00b7 rw [iteratedDerivWithin_dpsiTermC j (by omega) n hr]\n have hsm : r + n + 1 \u2260 0 := dpsiTermC_pole_free hr n\n apply DifferentiableAt.mul (differentiableAt_const _)\n have h4 : DifferentiableAt \u2102 (fun s : \u2102 => s + \u2191n + 1) r :=\n (differentiableAt_id.add (differentiableAt_const _)).add (differentiableAt_const _)\n exact (((hasDerivAt_zpow (-(j : \u2124) - 1) (r + \u2191n + 1) (Or.inl hsm)).differentiableAt).comp r\n h4.hasDerivAt)",
"new_string": " \u00b7 have h5 : iteratedDerivWithin j (fun z : \u2102 => dpsiTermC z n) dpsiDomain r =\n -((-1 : \u2102)) ^ j * (j ! : \u2102) * (r + \u2191n + 1) ^ (-(j : \u2124) - 1) :=\n iteratedDerivWithin_dpsiTermC j (by omega) n hr\n rw [h5]\n have hsm : r + n + 1 \u2260 0 := dpsiTermC_pole_free hr n\n apply DifferentiableAt.mul (differentiableAt_const _)\n have h4 : DifferentiableAt \u2102 (fun s : \u2102 => s + \u2191n + 1) r :=\n (differentiableAt_id.add (differentiableAt_const _)).add (differentiableAt_const _)\n exact (((hasDerivAt_zpow (-(j : \u2124) - 1) (r + \u2191n + 1) (Or.inl hsm)).differentiableAt).comp r\n h4.hasDerivAt)"
}
result 2026-08-22 08:52:42Z → 2026-08-23 11:15:24Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1.lean",
"oldString": " NguyenRH1.LiGapCircles \u2014 ii\u2032-5e-i: the \u03b3-gap circle radii\n (exists_gapCircle_radius): R \u2208 [4k+5/2,\n 4k+3] with dist(R/2, \u2124) \u2265 1/4 and every\n zero ordinate either \u2265 1 away or at\n clearance \u2265 1/(16\u00b7C0hat(4k+9/2)) \u2014 the\n \u00d72-rescaled midpoint pigeonhole",
"newString": " NguyenRH1.LiGapCircles \u2014 ii\u2032-5e-i: the \u03b3-gap circle radii\n (exists_gapCircle_radius): R \u2208 [4k+5/2,\n 4k+3] with dist(R/2, \u2124) \u2265 1/4 and every\n zero ordinate either \u2265 1 away or at\n clearance \u2265 1/(16\u00b7C0hat(4k+9/2)) \u2014 the\n \u00d72-rescaled midpoint pigeonhole\n NguyenRH1.LiDigammaLeft \u2014 ii\u2032-5e-ii (first half): the telescoping\n tail + the right-half-plane digamma bound\n (norm_digamma_le_of_two_le_re, Weierstrass\n split + harmonic + telescope) + the\n separated reciprocal sum bound\n (digamma_recip_sum_le, the V-split)",
"originalFile": "/-\nNguyenRH1.lean \u2014 root of the NGUYEN-RH-1 Phase-5 skeleton library.\n\nStatement-level Lean 4 skeletons for the project's new results\n(paper draft.tex \u00a73), built on zeta-23-lean (tag v1.0) conventions.\nEvery proof is `sorry` at this stage BY DESIGN: this is task 5.1's\nfirst pass \u2014 get the *statements* machine-checked against Mathlib +\nZeta23 types. `#print axioms` on any declaration here shows `sorryAx`\nuntil the corresponding proof is formalized; that marker is the\nhonesty tag, mirrored by the [SKELETON] comments.\n\nContents:\n NguyenRH1.LiSkeleton \u2014 R-004 finite-height Li theorem + the\n archimedean lower bound (draft \u00a73.1);\n assembly proved modulo SfBoundProp\n NguyenRH1.LiBasePoints \u2014 tier (ii) step 2: certified interval-bound\n discharge of the two margin evaluations;\n lambda_pos_finite_height_verified (perimeter\n + SfBoundProp only)\n NguyenRH1.LiResidue \u2014 tier ii\u2032-1b(i): the w = 1 residue of the\n Lagarias-kernel integrand (ResW1_eq_sum)\n NguyenRH1.LiPrincipal \u2014 tier ii\u2032-1b(ii): the principal-part integral\n (rectInt'_ppart)\n NguyenRH1.LiSides \u2014 tier ii\u2032-1b(iii): rectangle residue theorem\n for the integrand (rectInt'_Jlag)\n NguyenRH1.LiSwaps \u2014 tier ii\u2032-2: the elementary swaps\n (liCoeff_eq_sum_sigma, sInfty_eq_sum)\n NguyenRH1.LiZetaCoeff \u2014 tier ii\u2032-3a piece (B): the \u03b6-piece Taylor\n coefficients (iteratedDeriv_hLiExt)\n NguyenRH1.LiDigamma \u2014 tier ii\u2032-3a piece (A) core: the digamma\n series + polygamma values at 1/2\n NguyenRH1.LiArchimedean \u2014 tier ii\u2032-3a piece (A): the archimedean\n coefficients (iteratedDeriv_logDeriv_Gamma\u211d)\n NguyenRH1.LiContour \u2014 tier ii\u2032-4: L3 zero-avoiding T (pigeonhole)\n + L1 kernel bounds + zeroMult(\u22122) = 1\n NguyenRH1.LiRBound \u2014 tier ii\u2032-5b: the level-2 Euler\u2013Maclaurin \u03b6\n representation (re > \u22121) + the polynomial\n disc bound \u2016\u03b6\u2016 \u2264 (3/4)|t|\u00b2\n NguyenRH1.LiDisc \u2014 tier ii\u2032-5c: the loose Davenport-disc\n R-bound \u2016\u03b6\u2032/\u03b6 \u2212 \u03a3_near m/(s\u2212\u03c1)\u2016 \u2264\n 100000\u00b7(log|t|)\u00b2 on \u03c3 \u2208 [1/2, 7/2], |t| \u2265 10\n (Blaschke quotient + log + Borel\u2013Carath\u00e9odory)\n NguyenRH1.LiReflect \u2014 tier ii\u2032-5d: the FE-reflection of the loose\n R-bound to \u03c3 \u2208 [\u22125/2, 1/2] (exact zero-sum\n cancellation under \u03c1 \u21a6 1\u2212\u03c1; \u0393R-bracket via\n the +4 shift + digamma_stirling) + the\n left-plane uniform digamma bound\n \u2016\u03c8(w) \u2212 log w\u2016 \u2264 3 for |im w| \u2265 2\n NguyenRH1.LiTrivialZeros \u2014 ii\u2032-5e prep: zeroMult(\u22122n) = 1 for ALL\n n \u2265 1 (product-form \u0393 recurrence\n gamma_mul_prod + the HfunT cofactor;\n generalizes ii\u2032-4c's zeroMult_neg_two)\n NguyenRH1.LiGapCircles \u2014 ii\u2032-5e-i: the \u03b3-gap circle radii\n (exists_gapCircle_radius): R \u2208 [4k+5/2,\n 4k+3] with dist(R/2, \u2124) \u2265 1/4 and every\n zero ordinate either \u2265 1 away or at\n clearance \u2265 1/(16\u00b7C0hat(4k+9/2)) \u2014 the\n \u00d72-rescaled midpoint pigeonhole\n NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)\n NguyenRH1.ValleySkeleton \u2014 valley law statements (draft \u00a73.3)\n-/\nimport NguyenRH1.LiSkeleton\nimport NguyenRH1.LiBasePoints\nimport NguyenRH1.LiResidue\nimport NguyenRH1.LiPrincipal\nimport NguyenRH1.LiSides\nimport NguyenRH1.LiSwaps\nimport NguyenRH1.LiZetaCoeff\nimport NguyenRH1.LiDigamma\nimport NguyenRH1.LiArchimedean\nimport NguyenRH1.LiContour\nimport NguyenRH1.LiRBound\nimport NguyenRH1.LiDisc\nimport NguyenRH1.LiReflect\nimport NguyenRH1.LiTrivialZeros\nimport NguyenRH1.LiGapCircles\nimport NguyenRH1.SuzukiSkeleton\nimport NguyenRH1.ValleySkeleton\n",
"structuredPatch": [
{
"oldStart": 56,
"oldLines": 6,
"newStart": 56,
"newLines": 12,
"lines": [
" zero ordinate either \u2265 1 away or at",
" clearance \u2265 1/(16\u00b7C0hat(4k+9/2)) \u2014 the",
" \u00d72-rescaled midpoint pigeonhole",
"+ NguyenRH1.LiDigammaLeft \u2014 ii\u2032-5e-ii (first half): the telescoping",
"+ tail + the right-half-plane digamma bound",
"+ (norm_digamma_le_of_two_le_re, Weierstrass",
"+ split + harmonic + telescope) + the",
"+ separated reciprocal sum bound",
"+ (digamma_recip_sum_le, the V-split)",
" NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)",
" NguyenRH1.ValleySkeleton \u2014 valley law statements (draft \u00a73.3)",
" -/"
]
}
],
"userModified": false,
"replaceAll": false
}
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"old_string": " rw [show (-(k : \u2124) - 1 : \u2124) = -(((k + 1 : \u2115) : \u2124)) by push_cast; ring, zpow_neg, zpow_natCast]\n rw [div_eq_mul_inv]\n apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n rw [show (((m : \u211d) + 1) ^ (k + 1))\u207b\u00b9 = 1 / (((m : \u211d) + 1) ^ (k + 1)) from by rw [one_div]]\n rw [show (\u2016s + \u2191m + 1\u2016 ^ (k + 1))\u207b\u00b9 = 1 / (\u2016s + \u2191m + 1\u2016 ^ (k + 1)) from by rw [one_div]]\n apply one_div_le_one_div_of_le (by positivity)\n exact pow_le_pow_left\u2080 (by positivity : (0 : \u211d) \u2264 (m : \u211d) + 1) hnormS (k + 1)",
"new_string": " rw [show (-(k : \u2124) - 1 : \u2124) = -(((k + 1 : \u2115) : \u2124)) by push_cast; ring, zpow_neg, zpow_natCast]\n show (\u2191k ! : \u211d) * (\u2016s + \u2191m + 1\u2016 ^ (k + 1))\u207b\u00b9 \u2264 (k ! : \u211d) / ((m : \u211d) + 1) ^ (k + 1)\n rw [div_eq_mul_inv]\n apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n rw [\u2190 one_div, \u2190 one_div]\n apply one_div_le_one_div_of_le (by positivity)\n exact pow_le_pow_left\u2080 (by positivity : (0 : \u211d) \u2264 (m : \u211d) + 1) hnormS (k + 1)"
}
result 2026-08-22 08:52:42Z → 2026-08-23 11:16:18Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1.lean",
"oldString": "import NguyenRH1.LiGapCircles\nimport NguyenRH1.SuzukiSkeleton",
"newString": "import NguyenRH1.LiGapCircles\nimport NguyenRH1.LiDigammaLeft\nimport NguyenRH1.SuzukiSkeleton",
"originalFile": "/-\nNguyenRH1.lean \u2014 root of the NGUYEN-RH-1 Phase-5 skeleton library.\n\nStatement-level Lean 4 skeletons for the project's new results\n(paper draft.tex \u00a73), built on zeta-23-lean (tag v1.0) conventions.\nEvery proof is `sorry` at this stage BY DESIGN: this is task 5.1's\nfirst pass \u2014 get the *statements* machine-checked against Mathlib +\nZeta23 types. `#print axioms` on any declaration here shows `sorryAx`\nuntil the corresponding proof is formalized; that marker is the\nhonesty tag, mirrored by the [SKELETON] comments.\n\nContents:\n NguyenRH1.LiSkeleton \u2014 R-004 finite-height Li theorem + the\n archimedean lower bound (draft \u00a73.1);\n assembly proved modulo SfBoundProp\n NguyenRH1.LiBasePoints \u2014 tier (ii) step 2: certified interval-bound\n discharge of the two margin evaluations;\n lambda_pos_finite_height_verified (perimeter\n + SfBoundProp only)\n NguyenRH1.LiResidue \u2014 tier ii\u2032-1b(i): the w = 1 residue of the\n Lagarias-kernel integrand (ResW1_eq_sum)\n NguyenRH1.LiPrincipal \u2014 tier ii\u2032-1b(ii): the principal-part integral\n (rectInt'_ppart)\n NguyenRH1.LiSides \u2014 tier ii\u2032-1b(iii): rectangle residue theorem\n for the integrand (rectInt'_Jlag)\n NguyenRH1.LiSwaps \u2014 tier ii\u2032-2: the elementary swaps\n (liCoeff_eq_sum_sigma, sInfty_eq_sum)\n NguyenRH1.LiZetaCoeff \u2014 tier ii\u2032-3a piece (B): the \u03b6-piece Taylor\n coefficients (iteratedDeriv_hLiExt)\n NguyenRH1.LiDigamma \u2014 tier ii\u2032-3a piece (A) core: the digamma\n series + polygamma values at 1/2\n NguyenRH1.LiArchimedean \u2014 tier ii\u2032-3a piece (A): the archimedean\n coefficients (iteratedDeriv_logDeriv_Gamma\u211d)\n NguyenRH1.LiContour \u2014 tier ii\u2032-4: L3 zero-avoiding T (pigeonhole)\n + L1 kernel bounds + zeroMult(\u22122) = 1\n NguyenRH1.LiRBound \u2014 tier ii\u2032-5b: the level-2 Euler\u2013Maclaurin \u03b6\n representation (re > \u22121) + the polynomial\n disc bound \u2016\u03b6\u2016 \u2264 (3/4)|t|\u00b2\n NguyenRH1.LiDisc \u2014 tier ii\u2032-5c: the loose Davenport-disc\n R-bound \u2016\u03b6\u2032/\u03b6 \u2212 \u03a3_near m/(s\u2212\u03c1)\u2016 \u2264\n 100000\u00b7(log|t|)\u00b2 on \u03c3 \u2208 [1/2, 7/2], |t| \u2265 10\n (Blaschke quotient + log + Borel\u2013Carath\u00e9odory)\n NguyenRH1.LiReflect \u2014 tier ii\u2032-5d: the FE-reflection of the loose\n R-bound to \u03c3 \u2208 [\u22125/2, 1/2] (exact zero-sum\n cancellation under \u03c1 \u21a6 1\u2212\u03c1; \u0393R-bracket via\n the +4 shift + digamma_stirling) + the\n left-plane uniform digamma bound\n \u2016\u03c8(w) \u2212 log w\u2016 \u2264 3 for |im w| \u2265 2\n NguyenRH1.LiTrivialZeros \u2014 ii\u2032-5e prep: zeroMult(\u22122n) = 1 for ALL\n n \u2265 1 (product-form \u0393 recurrence\n gamma_mul_prod + the HfunT cofactor;\n generalizes ii\u2032-4c's zeroMult_neg_two)\n NguyenRH1.LiGapCircles \u2014 ii\u2032-5e-i: the \u03b3-gap circle radii\n (exists_gapCircle_radius): R \u2208 [4k+5/2,\n 4k+3] with dist(R/2, \u2124) \u2265 1/4 and every\n zero ordinate either \u2265 1 away or at\n clearance \u2265 1/(16\u00b7C0hat(4k+9/2)) \u2014 the\n \u00d72-rescaled midpoint pigeonhole\n NguyenRH1.LiDigammaLeft \u2014 ii\u2032-5e-ii (first half): the telescoping\n tail + the right-half-plane digamma bound\n (norm_digamma_le_of_two_le_re, Weierstrass\n split + harmonic + telescope) + the\n separated reciprocal sum bound\n (digamma_recip_sum_le, the V-split)\n NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)\n NguyenRH1.ValleySkeleton \u2014 valley law statements (draft \u00a73.3)\n-/\nimport NguyenRH1.LiSkeleton\nimport NguyenRH1.LiBasePoints\nimport NguyenRH1.LiResidue\nimport NguyenRH1.LiPrincipal\nimport NguyenRH1.LiSides\nimport NguyenRH1.LiSwaps\nimport NguyenRH1.LiZetaCoeff\nimport NguyenRH1.LiDigamma\nimport NguyenRH1.LiArchimedean\nimport NguyenRH1.LiContour\nimport NguyenRH1.LiRBound\nimport NguyenRH1.LiDisc\nimport NguyenRH1.LiReflect\nimport NguyenRH1.LiTrivialZeros\nimport NguyenRH1.LiGapCircles\nimport NguyenRH1.SuzukiSkeleton\nimport NguyenRH1.ValleySkeleton\n",
"structuredPatch": [
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"oldLines": 6,
"newStart": 80,
"newLines": 7,
"lines": [
" import NguyenRH1.LiReflect",
" import NguyenRH1.LiTrivialZeros",
" import NguyenRH1.LiGapCircles",
"+import NguyenRH1.LiDigammaLeft",
" import NguyenRH1.SuzukiSkeleton",
" import NguyenRH1.ValleySkeleton",
" "
]
}
],
"userModified": false,
"replaceAll": false
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raw content_json (database row)
[{"type":"thinking","thinking":"The code at 768 looks right. The issue must be the `by omega` in the rw argument: `iteratedDerivWithin_dpsiTermC j (by omega) n hr` — when used inside `rw [...]`, the `by omega` proof obligation: 1 ≤ j — with hj0 : j > 0 in context ✓ — hmm — actually, wait: the rcases at 757: `rcases Nat.eq_zero_or_pos j with hj0 | hj0` — in the second branch hj0 : 0 < j — and `(by omega : 1 ≤ j)` — omega with hj0 : 0 < j — 1 ≤ j ⟺ 0 < j ✓ should work...\n\nHmm — actually maybe the issue is different: the rw's term is elaborated with the `by omega` needing the goal... — when you write `rw [lemma (by omega) ...]`, the by-block is elaborated with the expected type 1 ≤ j — that's fine.\n\nLet me look at the error once more: \"Did not find an occurrence of the pattern iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r in the target expression DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r\" — the two ARE the same up to binder name... — UNLESS the pattern in the error display is what's generated from the lemma — and the target's — hmm — actually — Lean 4's rw DOES match up to alpha? Let me not assume — the classic behavior: rw matching is keyed by `kabstract` — which by default matches up to reducible defeq — and alpha-equivalence is NOT defeq (de Bruijn terms are equal but kabstract compares... — actually kabstract DOES handle alpha since de Bruijn makes binder names irrelevant — the displayed binder names are just pretty-printing). So alpha is not the issue.\n\nWAIT — I bet it's this: the lemma's conclusion LHS: `iteratedDerivWithin k (fun s : ℂ => dpsiTermC s m) dpsiDomain s` — but the GOAL's term: `iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain r` — with the DIFFERENCE: my lemma's function is `(fun s : ℂ => dpsiTermC s m)` — hmm same...\n\nHmm hold on — maybe the error is that `rw [iteratedDerivWithin_dpsiTermC j (by omega) n hr]` — the `by omega` inside a rw argument creates a NEW GOAL (the side condition isn't discharged inline)?? — no — `by omega` fully elaborates inline when the expected type is known (1 ≤ j).\n\nOK let me look at the actual error text once more, super carefully: \"Did not find an occurrence of the pattern `iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r` in the target expression `DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r`\" — hmm — IDENTICAL modulo binder. UNLESS — the `dpsiDomain` in the goal vs the lemma — dpsiDomain is a def — both reference the same def ✓. Or `iteratedDerivWithin` — same.\n\nLet me try a completely different approach: convert the goal first via `show`:\n```lean\n · have h4 : iteratedDerivWithin j (fun z : ℂ => dpsiTermC z n) dpsiDomain r =\n -((-1 : ℂ)) ^ j * (j ! : ℂ) * (r + ↑n + 1) ^ (-(j : ℤ) - 1) :=\n iteratedDerivWithin_dpsiTermC j (by omega) n hr\n rw [h4]\n ...\n```\nIf the application itself elaborates fine, the rw [h4] uses h4's LHS as the pattern — h4's LHS is written exactly as the goal's term ✓ (my text) — then it matches by construction. The earlier failure might be from how the lemma-application's elaboration normalized something. Let me use the explicit h4.","signature":""},{"type":"text","text":"The lemma application in rw is misbehaving — using an explicitly-typed `have` instead."},{"type":"tool_use","id":"Edit_432","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","old_string":" · rw [iteratedDerivWithin_dpsiTermC j (by omega) n hr]\n have hsm : r + n + 1 ≠ 0 := dpsiTermC_pole_free hr n\n apply DifferentiableAt.mul (differentiableAt_const _)\n have h4 : DifferentiableAt ℂ (fun s : ℂ => s + ↑n + 1) r :=\n (differentiableAt_id.add (differentiableAt_const _)).add (differentiableAt_const _)\n exact (((hasDerivAt_zpow (-(j : ℤ) - 1) (r + ↑n + 1) (Or.inl hsm)).differentiableAt).comp r\n h4.hasDerivAt)","new_string":" · have h5 : iteratedDerivWithin j (fun z : ℂ => dpsiTermC z n) dpsiDomain r =\n -((-1 : ℂ)) ^ j * (j ! : ℂ) * (r + ↑n + 1) ^ (-(j : ℤ) - 1) :=\n iteratedDerivWithin_dpsiTermC j (by omega) n hr\n rw [h5]\n have hsm : r + n + 1 ≠ 0 := dpsiTermC_pole_free hr n\n apply DifferentiableAt.mul (differentiableAt_const _)\n have h4 : DifferentiableAt ℂ (fun s : ℂ => s + ↑n + 1) r :=\n (differentiableAt_id.add (differentiableAt_const _)).add (differentiableAt_const _)\n exact (((hasDerivAt_zpow (-(j : ℤ) - 1) (r + ↑n + 1) (Or.inl hsm)).differentiableAt).comp r\n h4.hasDerivAt)"},"extra_content":null},{"type":"tool_use","id":"Edit_433","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","old_string":" rw [show (-(k : ℤ) - 1 : ℤ) = -(((k + 1 : ℕ) : ℤ)) by push_cast; ring, zpow_neg, zpow_natCast]\n rw [div_eq_mul_inv]\n apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n rw [show (((m : ℝ) + 1) ^ (k + 1))⁻¹ = 1 / (((m : ℝ) + 1) ^ (k + 1)) from by rw [one_div]]\n rw [show (‖s + ↑m + 1‖ ^ (k + 1))⁻¹ = 1 / (‖s + ↑m + 1‖ ^ (k + 1)) from by rw [one_div]]\n apply one_div_le_one_div_of_le (by positivity)\n exact pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (m : ℝ) + 1) hnormS (k + 1)","new_string":" rw [show (-(k : ℤ) - 1 : ℤ) = -(((k + 1 : ℕ) : ℤ)) by push_cast; ring, zpow_neg, zpow_natCast]\n show (↑k ! : ℝ) * (‖s + ↑m + 1‖ ^ (k + 1))⁻¹ ≤ (k ! : ℝ) / ((m : ℝ) + 1) ^ (k + 1)\n rw [div_eq_mul_inv]\n apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n rw [← one_div, ← one_div]\n apply one_div_le_one_div_of_le (by positivity)\n exact pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (m : ℝ) + 1) hnormS (k + 1)"},"extra_content":null}]