← Session 21 — Tier ii′ planning + launch: close SfBoundProp in Lean (D-064)
thinking (5412 chars)
No `ContinuousAt.sum` found. I'll prove tail continuity via `ContinuousAt.add` by Finset induction, or better — use `ContinuousAt.sum` alternative: `continuousAt_sum`? Not found either. Simplest robust: Finset induction:
```
lemma tail_continuousAt {n : ℕ} {w : ℂ} (hw : w ≠ 1) :
ContinuousAt (∑ i ∈ Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by
classical
refine Finset.induction (motive := fun s => ContinuousAt
(∑ i ∈ s, Acoeff n i / (w - 1) ^ i) w) ?_ ?_ (Finset.Icc 2 (n + 1))
· simp
· intro a s has hs
rw [Finset.sum_insert has]
apply ContinuousAt.add hs
· -- term continuity
have hw1 : w - 1 ≠ 0 := sub_ne_zero.mpr hw
apply ContinuousAt.div (continuousAt_const)
· exact (continuousAt_id.sub continuousAt_const).pow _
· exact pow_ne_zero _ hw1
```
Hmm — Finset.induction on (Icc 2 (n+1)) with motive — the `refine Finset.induction ?_ ?_ s` form: `Finset.induction (p : Finset α → Prop) (empty : p ∅) (insert : ∀ a s, a ∉ s → p s → p (insert a s)) : p s`? The argument order: `Finset.induction (h1 : p ∅) (h2 : ∀ ⦃a⦄ {s}, a ∉ s → p s → p (insert a s)) : p s` ✓. Let me write:
```
classical
refine Finset.induction (s := Finset.Icc 2 (n + 1)) ?_ ?_
· simp
· intro a s has hs
rw [Finset.sum_insert has]
exact ContinuousAt.add hs (term_cont _)
```
Hmm — motive inference might be tricky; alternative: `induction s using Finset.induction` — but the sum set is fixed (Icc 2 (n+1)) — induction on that Finset: `generalize` it first? Cleaner: use `Finset.induction_on`:
```
have : ∀ s : Finset ℕ, ContinuousAt (∑ i ∈ s, Acoeff n i / (w - 1) ^ i) w := by
intro s
induction s using Finset.induction with
| empty => simp
| insert has hs =>
rw [Finset.sum_insert has]
apply ContinuousAt.add hs
have hw1 : w - 1 ≠ 0 := sub_ne_zero.mpr hw
apply ContinuousAt.div (continuousAt_const)
· exact (continuousAt_id.sub continuousAt_const).pow _
· exact pow_ne_zero _ hw1
exact this _
```
Wait — ContinuousAt.div (hf) (hg) (h : g x ≠ 0) — signature check: `ContinuousAt.div (hf : ContinuousAt f x) (hg : ContinuousAt g x) (hz : g x ≠ 0) : ContinuousAt (f / g) x` ✓. And ContinuousAt.add (hf) (hg) ✓.
Hmm — but actually — the term `Acoeff n i / (w - 1) ^ i` — as a function of w: `fun w => Acoeff n i / (w - 1) ^ i` — ContinuousAt.div needs f and g as functions of w — with f := fun _ => Acoeff n i (const), g := fun w => (w−1)^i ✓.
Also `ContinuousAt.add` — signature: `ContinuousAt.add (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f + g) x` ✓ — with (f + g) w = f w + g w — the sum_insert gives ∑ + term ✓ Pi.add_apply defeq.
Let me now apply all the pending fixes to LiPrincipal.lean:
1. Acoeff factorial: `Nat.factorial (n + 1 - i)` explicit.
2. Gprim unfold: `unfold Gprim` — wait, in Gprim_differentiableAt and deriv_Gprim I have `rw [Gprim]` — replace with `unfold Gprim` — hmm unfold in tactic: `unfold Gprim` should work, or `simp only [Gprim]` — simp only with the def name unfolds via equation lemmas ✓ safer. Or `rw [Gprim]`→`simp only [Gprim]`. Hmm — simp only [Gprim] uses the generated equation lemma Gprim.eq_def — should work.
3. tail_continuousAt via induction.
4. hGint restructure with Gprim_differentiableOn_compl + DifferentiableOn.deriv.
5. Acoeff_one's rw [show ...] — with Nat.factorial explicit the `show n + 1 - 1 = n` rewrite should work (target `n + 1 - 1` appears in `Nat.factorial (n + 1 - 1)` and `iteratedDeriv (n + 1 - 1)`).
For Acoeff_one, also fix: rw [Acoeff, ResW1, show n + 1 - 1 = n from by omega] — after unfolding Acoeff/ResW1 (both have the same body structure), the remaining: iteratedDeriv n F 1 / Nat.factorial (n+1−1) = iteratedDeriv n F 1 / Nat.factorial n — the show rewrites n+1−1 → n ✓ then rfl. But rw [Acoeff] — same equation-lemma issue as Gprim? Use `simp only [Acoeff, ResW1]`? — the defs are plain defs: rw [Acoeff] should unfold via equation lemma... The earlier error at 164 was the factorial parse — with Nat.factorial explicit it might work. Let me use `unfold Acoeff ResW1` or simp only. I'll use `simp only [Acoeff, ResW1, show n + 1 - 1 = n from by omega]`? — simp only with a show-term — no, simp args are lemmas: `simp only [Acoeff, ResW1]` then the arithmetic `n + 1 - 1 = n`: simp can close via omega-normalization? `simp only [Acoeff, ResW1]` unfolds both, goal: iteratedDeriv (n+1−1) F 1 / Nat.factorial (n+1−1) = iteratedDeriv n F 1 / Nat.factorial n — then `congr 1`? the args n+1−1 vs n: `congr 1` gives n+1−1 = n and Nat.factorial (n+1−1) = Nat.factorial n — both by omega/rfl... `congr 1 <;> omega`? factorial one needs congr on the argument: `congr 2`? Let me just:
```
lemma Acoeff_one (n : ℕ) : Acoeff n 1 = ResW1 n := by
simp only [Acoeff, ResW1]
congr 1
· omega
· congr 1
omega
```
Hmm — congr 1 on `iteratedDeriv (n+1−1) F 1 / Nat.factorial (n+1−1) = iteratedDeriv n F 1 / Nat.factorial n` — splits into the numerators and denominators: numerator: iteratedDeriv (n+1−1) F 1 = iteratedDeriv n F 1 — congr: n+1−1 = n ✓ omega; denominator: Nat.factorial (n+1−1) = Nat.factorial n: congr 1: n+1−1 = n ✓ omega. Let me structure:
```
simp only [Acoeff, ResW1, show n + 1 - 1 = n from by omega]
```
simp only with the show-equation as a simp rule rewrites n+1−1 → n ✓ then rfl. simp only [X, Y, heq] where heq : n + 1 - 1 = n — used as a rewrite rule ✓ works.
Let me apply all fixes.
Applying all pending fixes with explicit factorial, simp-only unfolds, induction for tail continuity, and the hGint restructure.
tool: Bash
input
{
"command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. Acoeff: explicit Nat.factorial\ns = s.replace(\"\"\"def Acoeff (n i : \u2115) : \u2102 :=\n iteratedDeriv (n + 1 - i) (fun w : \u2102 => (w ^ n - (w - 1) ^ n) * hLi w) 1 / (n + 1 - i)!\"\"\",\n\"\"\"def Acoeff (n i : \u2115) : \u2102 :=\n iteratedDeriv (n + 1 - i) (fun w : \u2102 => (w ^ n - (w - 1) ^ n) * hLi w) 1 /\n Nat.factorial (n + 1 - i)\"\"\")\n\n# 2. Acoeff_one\ns = s.replace(\"\"\"lemma Acoeff_one (n : \u2115) : Acoeff n 1 = ResW1 n := by\n rw [Acoeff, ResW1, show n + 1 - 1 = n from by omega]\"\"\",\n\"\"\"lemma Acoeff_one (n : \u2115) : Acoeff n 1 = ResW1 n := by\n simp only [Acoeff, ResW1, show n + 1 - 1 = n from by omega]\"\"\")\n\n# 3. Gprim unfolds\ns = s.replace(\"\"\" rw [Gprim]\n apply DifferentiableAt.sum\"\"\",\n\"\"\" simp only [Gprim]\n apply DifferentiableAt.sum\"\"\")\ns = s.replace(\"\"\" rw [Gprim, deriv_sum (fun i hi => ?_)]\"\"\",\n\"\"\" simp only [Gprim]\n rw [deriv_sum (fun i hi => ?_)]\"\"\")\n\n# 4. tail_continuousAt via induction\ns = s.replace(\"\"\"lemma tail_continuousAt {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n ContinuousAt (\u2211 i \u2208 Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\n apply ContinuousAt.sum\n intro i hi\n rw [Finset.mem_Icc] at hi\n have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n apply ContinuousAt.div (continuousAt_const)\n \u00b7 exact (continuousAt_id.sub continuousAt_const).pow _\n \u00b7 exact pow_ne_zero _ hw1\"\"\",\n\"\"\"lemma tail_continuousAt {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n ContinuousAt (\u2211 i \u2208 Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\n have hterm : \u2200 i : \u2115, ContinuousAt (fun w => Acoeff n i / (w - 1) ^ i) w := by\n intro i\n have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n apply ContinuousAt.div (continuousAt_const)\n \u00b7 exact (continuousAt_id.sub continuousAt_const).pow _\n \u00b7 exact pow_ne_zero _ hw1\n have : \u2200 s : Finset \u2115, ContinuousAt (\u2211 i \u2208 s, Acoeff n i / (w - 1) ^ i) w := by\n intro s\n induction s using Finset.induction with\n | empty => simp\n | insert has hs =>\n rename_i a s\n rw [Finset.sum_insert has]\n exact ContinuousAt.add hs (hterm a)\n exact this _\"\"\")\n\n# 5. hGint restructure\ns = s.replace(\"\"\" have hGint : RectangleBorderIntegrable (deriv (Gprim n)) z w := by\n apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n intro s hs\n have hs1 : s \u2260 1 := hs.2\n rw [deriv_Gprim hs1]\n exact (tail_continuousAt hs1).differentiableAt.continuousWithinAt\"\"\",\n\"\"\" have hGint : RectangleBorderIntegrable (deriv (Gprim n)) z w := by\n apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n have hopen : DifferentiableOn \u2102 (deriv (Gprim n)) {1}\u1d9c := by\n have hd : DifferentiableOn \u2102 (Gprim n) {1}\u1d9c := by\n intro s hs\n exact (Gprim_differentiableAt (Set.mem_compl_singleton_iff.mp hs)).differentiableWithinAt\n exact hd.deriv isOpen_compl_singleton\n exact hopen.mono (Set.diff_subset_diff_left (Set.subset_univ _))\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
"description": "Apply assembly fixes and recompile",
"timeout": 420000
}
raw content_json (database row)
[{"type":"thinking","thinking":"No `ContinuousAt.sum` found. I'll prove tail continuity via `ContinuousAt.add` by Finset induction, or better — use `ContinuousAt.sum` alternative: `continuousAt_sum`? Not found either. Simplest robust: Finset induction:\n\n```\nlemma tail_continuousAt {n : ℕ} {w : ℂ} (hw : w ≠ 1) :\n ContinuousAt (∑ i ∈ Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\n classical\n refine Finset.induction (motive := fun s => ContinuousAt\n (∑ i ∈ s, Acoeff n i / (w - 1) ^ i) w) ?_ ?_ (Finset.Icc 2 (n + 1))\n · simp\n · intro a s has hs\n rw [Finset.sum_insert has]\n apply ContinuousAt.add hs\n · -- term continuity\n have hw1 : w - 1 ≠ 0 := sub_ne_zero.mpr hw\n apply ContinuousAt.div (continuousAt_const)\n · exact (continuousAt_id.sub continuousAt_const).pow _\n · exact pow_ne_zero _ hw1\n```\nHmm — Finset.induction on (Icc 2 (n+1)) with motive — the `refine Finset.induction ?_ ?_ s` form: `Finset.induction (p : Finset α → Prop) (empty : p ∅) (insert : ∀ a s, a ∉ s → p s → p (insert a s)) : p s`? The argument order: `Finset.induction (h1 : p ∅) (h2 : ∀ ⦃a⦄ {s}, a ∉ s → p s → p (insert a s)) : p s` ✓. Let me write:\n```\n classical\n refine Finset.induction (s := Finset.Icc 2 (n + 1)) ?_ ?_\n · simp\n · intro a s has hs\n rw [Finset.sum_insert has]\n exact ContinuousAt.add hs (term_cont _)\n```\nHmm — motive inference might be tricky; alternative: `induction s using Finset.induction` — but the sum set is fixed (Icc 2 (n+1)) — induction on that Finset: `generalize` it first? Cleaner: use `Finset.induction_on`:\n```\n have : ∀ s : Finset ℕ, ContinuousAt (∑ i ∈ s, Acoeff n i / (w - 1) ^ i) w := by\n intro s\n induction s using Finset.induction with\n | empty => simp\n | insert has hs =>\n rw [Finset.sum_insert has]\n apply ContinuousAt.add hs\n have hw1 : w - 1 ≠ 0 := sub_ne_zero.mpr hw\n apply ContinuousAt.div (continuousAt_const)\n · exact (continuousAt_id.sub continuousAt_const).pow _\n · exact pow_ne_zero _ hw1\n exact this _\n```\nWait — ContinuousAt.div (hf) (hg) (h : g x ≠ 0) — signature check: `ContinuousAt.div (hf : ContinuousAt f x) (hg : ContinuousAt g x) (hz : g x ≠ 0) : ContinuousAt (f / g) x` ✓. And ContinuousAt.add (hf) (hg) ✓.\n\nHmm — but actually — the term `Acoeff n i / (w - 1) ^ i` — as a function of w: `fun w => Acoeff n i / (w - 1) ^ i` — ContinuousAt.div needs f and g as functions of w — with f := fun _ => Acoeff n i (const), g := fun w => (w−1)^i ✓.\n\nAlso `ContinuousAt.add` — signature: `ContinuousAt.add (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f + g) x` ✓ — with (f + g) w = f w + g w — the sum_insert gives ∑ + term ✓ Pi.add_apply defeq.\n\nLet me now apply all the pending fixes to LiPrincipal.lean:\n1. Acoeff factorial: `Nat.factorial (n + 1 - i)` explicit.\n2. Gprim unfold: `unfold Gprim` — wait, in Gprim_differentiableAt and deriv_Gprim I have `rw [Gprim]` — replace with `unfold Gprim` — hmm unfold in tactic: `unfold Gprim` should work, or `simp only [Gprim]` — simp only with the def name unfolds via equation lemmas ✓ safer. Or `rw [Gprim]`→`simp only [Gprim]`. Hmm — simp only [Gprim] uses the generated equation lemma Gprim.eq_def — should work.\n3. tail_continuousAt via induction.\n4. hGint restructure with Gprim_differentiableOn_compl + DifferentiableOn.deriv.\n5. Acoeff_one's rw [show ...] — with Nat.factorial explicit the `show n + 1 - 1 = n` rewrite should work (target `n + 1 - 1` appears in `Nat.factorial (n + 1 - 1)` and `iteratedDeriv (n + 1 - 1)`).\n\nFor Acoeff_one, also fix: rw [Acoeff, ResW1, show n + 1 - 1 = n from by omega] — after unfolding Acoeff/ResW1 (both have the same body structure), the remaining: iteratedDeriv n F 1 / Nat.factorial (n+1−1) = iteratedDeriv n F 1 / Nat.factorial n — the show rewrites n+1−1 → n ✓ then rfl. But rw [Acoeff] — same equation-lemma issue as Gprim? Use `simp only [Acoeff, ResW1]`? — the defs are plain defs: rw [Acoeff] should unfold via equation lemma... The earlier error at 164 was the factorial parse — with Nat.factorial explicit it might work. Let me use `unfold Acoeff ResW1` or simp only. I'll use `simp only [Acoeff, ResW1, show n + 1 - 1 = n from by omega]`? — simp only with a show-term — no, simp args are lemmas: `simp only [Acoeff, ResW1]` then the arithmetic `n + 1 - 1 = n`: simp can close via omega-normalization? `simp only [Acoeff, ResW1]` unfolds both, goal: iteratedDeriv (n+1−1) F 1 / Nat.factorial (n+1−1) = iteratedDeriv n F 1 / Nat.factorial n — then `congr 1`? the args n+1−1 vs n: `congr 1` gives n+1−1 = n and Nat.factorial (n+1−1) = Nat.factorial n — both by omega/rfl... `congr 1 <;> omega`? factorial one needs congr on the argument: `congr 2`? Let me just:\n```\nlemma Acoeff_one (n : ℕ) : Acoeff n 1 = ResW1 n := by\n simp only [Acoeff, ResW1]\n congr 1\n · omega\n · congr 1\n omega\n```\nHmm — congr 1 on `iteratedDeriv (n+1−1) F 1 / Nat.factorial (n+1−1) = iteratedDeriv n F 1 / Nat.factorial n` — splits into the numerators and denominators: numerator: iteratedDeriv (n+1−1) F 1 = iteratedDeriv n F 1 — congr: n+1−1 = n ✓ omega; denominator: Nat.factorial (n+1−1) = Nat.factorial n: congr 1: n+1−1 = n ✓ omega. Let me structure:\n```\n simp only [Acoeff, ResW1, show n + 1 - 1 = n from by omega]\n```\nsimp only with the show-equation as a simp rule rewrites n+1−1 → n ✓ then rfl. simp only [X, Y, heq] where heq : n + 1 - 1 = n — used as a rewrite rule ✓ works.\n\nLet me apply all fixes.","signature":""},{"type":"text","text":"Applying all pending fixes with explicit factorial, simp-only unfolds, induction for tail continuity, and the hGint restructure."},{"type":"tool_use","id":"Bash_267","name":"Bash","input":{"command":"cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. Acoeff: explicit Nat.factorial\ns = s.replace(\"\"\"def Acoeff (n i : ℕ) : ℂ :=\n iteratedDeriv (n + 1 - i) (fun w : ℂ => (w ^ n - (w - 1) ^ n) * hLi w) 1 / (n + 1 - i)!\"\"\",\n\"\"\"def Acoeff (n i : ℕ) : ℂ :=\n iteratedDeriv (n + 1 - i) (fun w : ℂ => (w ^ n - (w - 1) ^ n) * hLi w) 1 /\n Nat.factorial (n + 1 - i)\"\"\")\n\n# 2. Acoeff_one\ns = s.replace(\"\"\"lemma Acoeff_one (n : ℕ) : Acoeff n 1 = ResW1 n := by\n rw [Acoeff, ResW1, show n + 1 - 1 = n from by omega]\"\"\",\n\"\"\"lemma Acoeff_one (n : ℕ) : Acoeff n 1 = ResW1 n := by\n simp only [Acoeff, ResW1, show n + 1 - 1 = n from by omega]\"\"\")\n\n# 3. Gprim unfolds\ns = s.replace(\"\"\" rw [Gprim]\n apply DifferentiableAt.sum\"\"\",\n\"\"\" simp only [Gprim]\n apply DifferentiableAt.sum\"\"\")\ns = s.replace(\"\"\" rw [Gprim, deriv_sum (fun i hi => ?_)]\"\"\",\n\"\"\" simp only [Gprim]\n rw [deriv_sum (fun i hi => ?_)]\"\"\")\n\n# 4. tail_continuousAt via induction\ns = s.replace(\"\"\"lemma tail_continuousAt {n : ℕ} {w : ℂ} (hw : w ≠ 1) :\n ContinuousAt (∑ i ∈ Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\n apply ContinuousAt.sum\n intro i hi\n rw [Finset.mem_Icc] at hi\n have hw1 : w - 1 ≠ 0 := sub_ne_zero.mpr hw\n apply ContinuousAt.div (continuousAt_const)\n · exact (continuousAt_id.sub continuousAt_const).pow _\n · exact pow_ne_zero _ hw1\"\"\",\n\"\"\"lemma tail_continuousAt {n : ℕ} {w : ℂ} (hw : w ≠ 1) :\n ContinuousAt (∑ i ∈ Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\n have hterm : ∀ i : ℕ, ContinuousAt (fun w => Acoeff n i / (w - 1) ^ i) w := by\n intro i\n have hw1 : w - 1 ≠ 0 := sub_ne_zero.mpr hw\n apply ContinuousAt.div (continuousAt_const)\n · exact (continuousAt_id.sub continuousAt_const).pow _\n · exact pow_ne_zero _ hw1\n have : ∀ s : Finset ℕ, ContinuousAt (∑ i ∈ s, Acoeff n i / (w - 1) ^ i) w := by\n intro s\n induction s using Finset.induction with\n | empty => simp\n | insert has hs =>\n rename_i a s\n rw [Finset.sum_insert has]\n exact ContinuousAt.add hs (hterm a)\n exact this _\"\"\")\n\n# 5. hGint restructure\ns = s.replace(\"\"\" have hGint : RectangleBorderIntegrable (deriv (Gprim n)) z w := by\n apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n intro s hs\n have hs1 : s ≠ 1 := hs.2\n rw [deriv_Gprim hs1]\n exact (tail_continuousAt hs1).differentiableAt.continuousWithinAt\"\"\",\n\"\"\" have hGint : RectangleBorderIntegrable (deriv (Gprim n)) z w := by\n apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n have hopen : DifferentiableOn ℂ (deriv (Gprim n)) {1}ᶜ := by\n have hd : DifferentiableOn ℂ (Gprim n) {1}ᶜ := by\n intro s hs\n exact (Gprim_differentiableAt (Set.mem_compl_singleton_iff.mp hs)).differentiableWithinAt\n exact hd.deriv isOpen_compl_singleton\n exact hopen.mono (Set.diff_subset_diff_left (Set.subset_univ _))\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40","description":"Apply assembly fixes and recompile","timeout":420000},"extra_content":null}]