← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (7097 chars)
Two errors:
1. **Line 990 (hcont)**: `ContinuousAt.div` needs a denominator-nonzero side condition — the form: ContinuousAt.div (hf) (hg : ContinuousAt g x) (h : g x ≠ 0)? — my `(continuousAt_id.add continuousAt_const).div continuousAt_const` — the const-2 denominator — ContinuousAt.div requires g x ≠ 0 — for g := fun _ => 2: 2 ≠ 0 — add it: `.div continuousAt_const (by norm_num)` — hmm the signature: ContinuousAt.div (hf : ContinuousAt f x) (hg : ContinuousAt g x) (h : g x ≠ 0)? — so `((continuousAt_id.add continuousAt_const).div continuousAt_const (by norm_num))` — the last arg the ≠ 0 proof ✓. Also the resulting function is `((id + const) / const 2)` — the Pi forms vs my lambda fun u => (1+u)/2 — hmm — the div gives (id + fun _ => 1) / (fun _ => 2) — Pi.div — vs (fun u => (1+u)/2) — defeq-ish — the expected type (from my `have hc : ContinuousAt (fun u : ℂ => (1+u)/2) t`) — the div's function: ((id + fun _ => 1) / fun _ => 2) — applied u: (u + 1)/2 — vs (1+u)/2 — comm! — id + const = u + 1 not 1 + u!! — the association/comm — (hasDerivAt_const...).add (hasDerivAt_id) vs (hasDerivAt_id).add_const — the order: I wrote `(continuousAt_id.add continuousAt_const)` — id + const — (u + 1) — my target (1 + u) — comm — mismatch. Use `(continuousAt_const.add continuousAt_id)` — const + id = 1 + u ✓.
Let me fix: `((continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num))` — the function: ((fun _ => 1) + id) / (fun _ => 2) — applied: (1 + u)/2 ✓ matches my stated (1+u)/2 ✓.
2. **Line 1001**: `rw [← iteratedDeriv_succ]` — pattern `deriv (iteratedDeriv ?m ?m)` not found in the HasDerivAt goal — the goal: HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j+1) F (...)) (...) — iteratedDeriv_succ : iteratedDeriv (n+1) f = deriv (iteratedDeriv n f) — the ← direction: deriv (iteratedDeriv n f) → iteratedDeriv (n+1) f — the pattern deriv (iteratedDeriv ?n ?f) — the goal's derivative-value is iteratedDeriv (j+1) F ... — not of the deriv-form — and the FUNCTION is iteratedDeriv j F — hmm — I want to prove HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j+1) F pt) pt — by definition iteratedDeriv (j+1) F = deriv (iteratedDeriv j F) (succ) — so the value IS deriv (iteratedDeriv j F) pt — so the goal is HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) pt) pt — provable by DifferentiableAt.hasDerivAt of (analyticAt_iteratedDeriv hF j).deriv.differentiableAt — the iteratedDeriv j F is analytic at pt (analyticAt_iteratedDeriv) — its deriv is analytic at pt (AnalyticAt.deriv) — so DifferentiableAt (deriv (iteratedDeriv j F)) pt ✓ — but the goal's derivative VALUE is iteratedDeriv (j+1) F pt — and deriv (iteratedDeriv j F) pt = iteratedDeriv (j+1) F pt via iteratedDeriv_succ ✓ — so:
```lean
have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))
((1 + s) / 2) := by
have h9 : AnalyticAt ℂ (iteratedDeriv j F) ((1 + s) / 2) := analyticAt_iteratedDeriv hF j
have h10 : HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) ((1 + s) / 2))
((1 + s) / 2) := h9.differentiableAt.hasDerivAt
rwa [iteratedDeriv_succ] at h10 -- hmm — iteratedDeriv_succ : iteratedDeriv (j+1) F = deriv (iteratedDeriv j F) — rw at h10 rewrites the VALUE iteratedDeriv (j+1) F pt → deriv (iteratedDeriv j F) pt?? no — I want the reverse: h10's value deriv (iteratedDeriv j F) pt = iteratedDeriv (j+1) F pt — rw [iteratedDeriv_succ] at h10 would rewrite iteratedDeriv (j+1) F → deriv (iteratedDeriv j F) — but h10 has deriv (iteratedDeriv j F) — the pattern iteratedDeriv (j+1) F isn't in h10 — so rw fails. Use: rw [show iteratedDeriv (j + 1) F ((1+s)/2) = deriv (iteratedDeriv j F) ((1+s)/2) from congr_fun (iteratedDeriv_succ) _]?? — iteratedDeriv_succ : iteratedDeriv (n+1) f = deriv (iteratedDeriv n f) — as a function equality: congr_fun (iteratedDeriv_succ) ((1+s)/2) : iteratedDeriv (j+1) F ((1+s)/2) = deriv (iteratedDeriv j F) ((1+s)/2) ✓ — so `rw [congr_fun (iteratedDeriv_succ) ((1+s)/2)]`?? on the GOAL — goal HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j+1) F pt) pt — rw [that] rewrites the value → deriv (iteratedDeriv j F) pt ✓ then exact h10 ✓. Hmm — iteratedDeriv_succ's implicit n f — congr_fun (iteratedDeriv_succ) — the n f implicit — from the target... Let me write it as:
have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))
((1 + s) / 2) := by
rw [iteratedDeriv_succ]
exact (analyticAt_iteratedDeriv hF j).deriv.differentiableAt.hasDerivAt
```
rw [iteratedDeriv_succ] on the GOAL: the goal's value `iteratedDeriv (j + 1) F ((1+s)/2)` — iteratedDeriv_succ's LHS pattern iteratedDeriv (?n+1) ?f — matches with n := j, f := F — rewrites the FUNCTION iteratedDeriv (j+1) F → deriv (iteratedDeriv j F) — the goal's value is the APPLICATION iteratedDeriv (j+1) F ((1+s)/2) — the pattern iteratedDeriv (n+1) f — the goal's term is (iteratedDeriv (j+1) F) ((1+s)/2) — an application of the function — the pattern iteratedDeriv (?n+1) ?f as a subterm — the goal has it (the function part of the application) ✓ rw rewrites the function part → deriv (iteratedDeriv j F) — the goal becomes HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) ((1+s)/2)) ((1+s)/2) ✓ then exact (analyticAt_iteratedDeriv hF j).deriv.differentiableAt.hasDerivAt ✓ — wait — but the deriv (iteratedDeriv j F) — AnalyticAt.deriv gives AnalyticAt (deriv (iteratedDeriv j F)) — then differentiableAt — hasDerivAt : HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) pt) pt ✓ — yes — the `X.deriv.differentiableAt.hasDerivAt` — (analyticAt_iteratedDeriv hF j) : AnalyticAt (iteratedDeriv j F) pt — .deriv : AnalyticAt (deriv (iteratedDeriv j F)) pt — .differentiableAt : DifferentiableAt (deriv ...) pt — .hasDerivAt : HasDerivAt (deriv ...) (deriv (deriv ...) pt) pt?!?! — WRONG — hasDerivAt of the DERIV function gives HasDerivAt (deriv (iteratedDeriv j F)) (deriv (deriv (iteratedDeriv j F)) pt) pt — that's the derivative OF the derivative — not what I want. I want: HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) pt) pt — that's just (analyticAt_iteratedDeriv hF j).differentiableAt.hasDerivAt — DifferentiableAt.hasDerivAt : DifferentiableAt f x → HasDerivAt f (deriv f x) x ✓ — so:
```lean
have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))
((1 + s) / 2) := by
rw [iteratedDeriv_succ]
exact (analyticAt_iteratedDeriv hF j).differentiableAt.hasDerivAt
```
✓ — the rw on the goal then exact — the goal after rw: HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) pt) pt — exact (AnalyticAt-iteratedDeriv).differentiableAt.hasDerivAt ✓.
Hmm wait — does rw [iteratedDeriv_succ] on the goal work? — goal: HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j+1) F pt) pt — the pattern iteratedDeriv (?n+1) ?f — occurs in the VALUE (iteratedDeriv (j+1) F pt — the application (iteratedDeriv (j+1) F) pt — the subterm iteratedDeriv (j+1) F ✓ matches) — rewrite → deriv (iteratedDeriv j F) — the value becomes (deriv (iteratedDeriv j F)) pt ✓ then exact ✓.
Two fixes: the `ContinuousAt.div` side condition/order, and the deriv-of-iteratedDeriv direction.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"old_string": " have hcont : Tendsto (fun u : \u2102 => (1 + u) / 2) (\ud835\udcdd t) (\ud835\udcdd ((1 + t) / 2)) := by\n have hc : ContinuousAt (fun u : \u2102 => (1 + u) / 2) t :=\n (continuousAt_id.add continuousAt_const).div continuousAt_const\n exact hc.tendsto\n exact hcont.eventually ht.eventually_analyticAt",
"new_string": " have hcont : Tendsto (fun u : \u2102 => (1 + u) / 2) (\ud835\udcdd t) (\ud835\udcdd ((1 + t) / 2)) := by\n have hc : ContinuousAt (fun u : \u2102 => (1 + u) / 2) t :=\n (continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num)\n exact hc.tendsto\n exact hcont.eventually ht.eventually_analyticAt"
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"old_string": " have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))\n ((1 + s) / 2) := by\n rw [\u2190 iteratedDeriv_succ]\n exact (analyticAt_iteratedDeriv hF j).deriv.differentiableAt.hasDerivAt",
"new_string": " have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))\n ((1 + s) / 2) := by\n rw [iteratedDeriv_succ]\n exact (analyticAt_iteratedDeriv hF j).differentiableAt.hasDerivAt"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Two errors:\n\n1. **Line 990 (hcont)**: `ContinuousAt.div` needs a denominator-nonzero side condition — the form: ContinuousAt.div (hf) (hg : ContinuousAt g x) (h : g x ≠ 0)? — my `(continuousAt_id.add continuousAt_const).div continuousAt_const` — the const-2 denominator — ContinuousAt.div requires g x ≠ 0 — for g := fun _ => 2: 2 ≠ 0 — add it: `.div continuousAt_const (by norm_num)` — hmm the signature: ContinuousAt.div (hf : ContinuousAt f x) (hg : ContinuousAt g x) (h : g x ≠ 0)? — so `((continuousAt_id.add continuousAt_const).div continuousAt_const (by norm_num))` — the last arg the ≠ 0 proof ✓. Also the resulting function is `((id + const) / const 2)` — the Pi forms vs my lambda fun u => (1+u)/2 — hmm — the div gives (id + fun _ => 1) / (fun _ => 2) — Pi.div — vs (fun u => (1+u)/2) — defeq-ish — the expected type (from my `have hc : ContinuousAt (fun u : ℂ => (1+u)/2) t`) — the div's function: ((id + fun _ => 1) / fun _ => 2) — applied u: (u + 1)/2 — vs (1+u)/2 — comm! — id + const = u + 1 not 1 + u!! — the association/comm — (hasDerivAt_const...).add (hasDerivAt_id) vs (hasDerivAt_id).add_const — the order: I wrote `(continuousAt_id.add continuousAt_const)` — id + const — (u + 1) — my target (1 + u) — comm — mismatch. Use `(continuousAt_const.add continuousAt_id)` — const + id = 1 + u ✓.\n\nLet me fix: `((continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num))` — the function: ((fun _ => 1) + id) / (fun _ => 2) — applied: (1 + u)/2 ✓ matches my stated (1+u)/2 ✓.\n\n2. **Line 1001**: `rw [← iteratedDeriv_succ]` — pattern `deriv (iteratedDeriv ?m ?m)` not found in the HasDerivAt goal — the goal: HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j+1) F (...)) (...) — iteratedDeriv_succ : iteratedDeriv (n+1) f = deriv (iteratedDeriv n f) — the ← direction: deriv (iteratedDeriv n f) → iteratedDeriv (n+1) f — the pattern deriv (iteratedDeriv ?n ?f) — the goal's derivative-value is iteratedDeriv (j+1) F ... — not of the deriv-form — and the FUNCTION is iteratedDeriv j F — hmm — I want to prove HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j+1) F pt) pt — by definition iteratedDeriv (j+1) F = deriv (iteratedDeriv j F) (succ) — so the value IS deriv (iteratedDeriv j F) pt — so the goal is HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) pt) pt — provable by DifferentiableAt.hasDerivAt of (analyticAt_iteratedDeriv hF j).deriv.differentiableAt — the iteratedDeriv j F is analytic at pt (analyticAt_iteratedDeriv) — its deriv is analytic at pt (AnalyticAt.deriv) — so DifferentiableAt (deriv (iteratedDeriv j F)) pt ✓ — but the goal's derivative VALUE is iteratedDeriv (j+1) F pt — and deriv (iteratedDeriv j F) pt = iteratedDeriv (j+1) F pt via iteratedDeriv_succ ✓ — so:\n```lean\n have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))\n ((1 + s) / 2) := by\n have h9 : AnalyticAt ℂ (iteratedDeriv j F) ((1 + s) / 2) := analyticAt_iteratedDeriv hF j\n have h10 : HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) ((1 + s) / 2))\n ((1 + s) / 2) := h9.differentiableAt.hasDerivAt\n rwa [iteratedDeriv_succ] at h10 -- hmm — iteratedDeriv_succ : iteratedDeriv (j+1) F = deriv (iteratedDeriv j F) — rw at h10 rewrites the VALUE iteratedDeriv (j+1) F pt → deriv (iteratedDeriv j F) pt?? no — I want the reverse: h10's value deriv (iteratedDeriv j F) pt = iteratedDeriv (j+1) F pt — rw [iteratedDeriv_succ] at h10 would rewrite iteratedDeriv (j+1) F → deriv (iteratedDeriv j F) — but h10 has deriv (iteratedDeriv j F) — the pattern iteratedDeriv (j+1) F isn't in h10 — so rw fails. Use: rw [show iteratedDeriv (j + 1) F ((1+s)/2) = deriv (iteratedDeriv j F) ((1+s)/2) from congr_fun (iteratedDeriv_succ) _]?? — iteratedDeriv_succ : iteratedDeriv (n+1) f = deriv (iteratedDeriv n f) — as a function equality: congr_fun (iteratedDeriv_succ) ((1+s)/2) : iteratedDeriv (j+1) F ((1+s)/2) = deriv (iteratedDeriv j F) ((1+s)/2) ✓ — so `rw [congr_fun (iteratedDeriv_succ) ((1+s)/2)]`?? on the GOAL — goal HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j+1) F pt) pt — rw [that] rewrites the value → deriv (iteratedDeriv j F) pt ✓ then exact h10 ✓. Hmm — iteratedDeriv_succ's implicit n f — congr_fun (iteratedDeriv_succ) — the n f implicit — from the target... Let me write it as:\n have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))\n ((1 + s) / 2) := by\n rw [iteratedDeriv_succ]\n exact (analyticAt_iteratedDeriv hF j).deriv.differentiableAt.hasDerivAt\n```\nrw [iteratedDeriv_succ] on the GOAL: the goal's value `iteratedDeriv (j + 1) F ((1+s)/2)` — iteratedDeriv_succ's LHS pattern iteratedDeriv (?n+1) ?f — matches with n := j, f := F — rewrites the FUNCTION iteratedDeriv (j+1) F → deriv (iteratedDeriv j F) — the goal's value is the APPLICATION iteratedDeriv (j+1) F ((1+s)/2) — the pattern iteratedDeriv (n+1) f — the goal's term is (iteratedDeriv (j+1) F) ((1+s)/2) — an application of the function — the pattern iteratedDeriv (?n+1) ?f as a subterm — the goal has it (the function part of the application) ✓ rw rewrites the function part → deriv (iteratedDeriv j F) — the goal becomes HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) ((1+s)/2)) ((1+s)/2) ✓ then exact (analyticAt_iteratedDeriv hF j).deriv.differentiableAt.hasDerivAt ✓ — wait — but the deriv (iteratedDeriv j F) — AnalyticAt.deriv gives AnalyticAt (deriv (iteratedDeriv j F)) — then differentiableAt — hasDerivAt : HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) pt) pt ✓ — yes — the `X.deriv.differentiableAt.hasDerivAt` — (analyticAt_iteratedDeriv hF j) : AnalyticAt (iteratedDeriv j F) pt — .deriv : AnalyticAt (deriv (iteratedDeriv j F)) pt — .differentiableAt : DifferentiableAt (deriv ...) pt — .hasDerivAt : HasDerivAt (deriv ...) (deriv (deriv ...) pt) pt?!?! — WRONG — hasDerivAt of the DERIV function gives HasDerivAt (deriv (iteratedDeriv j F)) (deriv (deriv (iteratedDeriv j F)) pt) pt — that's the derivative OF the derivative — not what I want. I want: HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) pt) pt — that's just (analyticAt_iteratedDeriv hF j).differentiableAt.hasDerivAt — DifferentiableAt.hasDerivAt : DifferentiableAt f x → HasDerivAt f (deriv f x) x ✓ — so:\n```lean\n have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))\n ((1 + s) / 2) := by\n rw [iteratedDeriv_succ]\n exact (analyticAt_iteratedDeriv hF j).differentiableAt.hasDerivAt\n```\n✓ — the rw on the goal then exact — the goal after rw: HasDerivAt (iteratedDeriv j F) (deriv (iteratedDeriv j F) pt) pt — exact (AnalyticAt-iteratedDeriv).differentiableAt.hasDerivAt ✓.\n\nHmm wait — does rw [iteratedDeriv_succ] on the goal work? — goal: HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j+1) F pt) pt — the pattern iteratedDeriv (?n+1) ?f — occurs in the VALUE (iteratedDeriv (j+1) F pt — the application (iteratedDeriv (j+1) F) pt — the subterm iteratedDeriv (j+1) F ✓ matches) — rewrite → deriv (iteratedDeriv j F) — the value becomes (deriv (iteratedDeriv j F)) pt ✓ then exact ✓.","signature":""},{"type":"text","text":"Two fixes: the `ContinuousAt.div` side condition/order, and the deriv-of-iteratedDeriv direction."},{"type":"tool_use","id":"Edit_531","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","old_string":" have hcont : Tendsto (fun u : ℂ => (1 + u) / 2) (𝓝 t) (𝓝 ((1 + t) / 2)) := by\n have hc : ContinuousAt (fun u : ℂ => (1 + u) / 2) t :=\n (continuousAt_id.add continuousAt_const).div continuousAt_const\n exact hc.tendsto\n exact hcont.eventually ht.eventually_analyticAt","new_string":" have hcont : Tendsto (fun u : ℂ => (1 + u) / 2) (𝓝 t) (𝓝 ((1 + t) / 2)) := by\n have hc : ContinuousAt (fun u : ℂ => (1 + u) / 2) t :=\n (continuousAt_const.add continuousAt_id).div continuousAt_const (by norm_num)\n exact hc.tendsto\n exact hcont.eventually ht.eventually_analyticAt"},"extra_content":null},{"type":"tool_use","id":"Edit_532","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","old_string":" have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))\n ((1 + s) / 2) := by\n rw [← iteratedDeriv_succ]\n exact (analyticAt_iteratedDeriv hF j).deriv.differentiableAt.hasDerivAt","new_string":" have hd : HasDerivAt (iteratedDeriv j F) (iteratedDeriv (j + 1) F ((1 + s) / 2))\n ((1 + s) / 2) := by\n rw [iteratedDeriv_succ]\n exact (analyticAt_iteratedDeriv hF j).differentiableAt.hasDerivAt"},"extra_content":null}]