← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (1480 chars)
The comp problem: `(analyticAt_digamma h).comp h1` — analyticAt_digamma h : AnalyticAt ℂ digamma (s / 2). AnalyticAt.comp signature: `theorem AnalyticAt.comp {f : E → F} {g : F → G} {x : E} (hg : AnalyticAt 𝕜 g (f x)) (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (g ∘ f) x` — so hg's type AnalyticAt ℂ digamma (s/2) must unify with AnalyticAt ℂ g (f x): g := digamma, f x := s/2 — with f and x as metavars: the pattern `f x` is a HIGHER-ORDER unification problem: f x =?= s/2 — HOU: f := fun s => s/2, x := s is one solution; f := fun _ => s/2, x := anything another... The elaborator picked f := HDiv.hDiv s (function fun y => y/2? no — `HDiv.hDiv s` = fun y => s / y?? HDiv.hDiv s : ℂ → ℂ = fun y => HDiv.hDiv s y = s/y — WRONG branch of HOU) and x := 2: f x = s/2 ✓ that's the other solution! Classic HOU ambiguity — the elaborator chose f := HDiv.hDiv s, x := 2.
Fix: pin f explicitly: `AnalyticAt.comp (f := fun s : ℂ => s / 2) (g := Complex.digamma) (analyticAt_digamma h) h1` — wait the named args: AnalyticAt.comp (hg) (hf) with {f g x} implicit: provide (f := fun s : ℂ => s / 2): then hg : AnalyticAt ℂ digamma ((fun s => s/2) x) — with x to unify: hg := analyticAt_digamma h : AnalyticAt ℂ digamma (s/2): (fun s => s/2) x =?= s/2: x := s ✓ (beta HOU with f pinned is first-order ✓). Then hf := h1 : AnalyticAt ℂ (fun s => s/2) s ✓.
So: `have h2 : AnalyticAt ℂ (Complex.digamma ∘ fun s : ℂ => s / 2) s := AnalyticAt.comp (f := fun s : ℂ => s / 2) (analyticAt_digamma h) h1`.
Classic higher-order unification ambiguity in `AnalyticAt.comp` — the elaborator picked `f := HDiv.hDiv s, x := 2`. Pinning `f` explicitly.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean",
"old_string": " have h2 : AnalyticAt \u2102 (Complex.digamma \u2218 fun s : \u2102 => s / 2) s :=\n (analyticAt_digamma h).comp h1",
"new_string": " have h2 : AnalyticAt \u2102 (Complex.digamma \u2218 fun s : \u2102 => s / 2) s :=\n AnalyticAt.comp (f := fun s : \u2102 => s / 2) (analyticAt_digamma h) h1"
}
raw content_json (database row)
[{"type":"thinking","thinking":"The comp problem: `(analyticAt_digamma h).comp h1` — analyticAt_digamma h : AnalyticAt ℂ digamma (s / 2). AnalyticAt.comp signature: `theorem AnalyticAt.comp {f : E → F} {g : F → G} {x : E} (hg : AnalyticAt 𝕜 g (f x)) (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (g ∘ f) x` — so hg's type AnalyticAt ℂ digamma (s/2) must unify with AnalyticAt ℂ g (f x): g := digamma, f x := s/2 — with f and x as metavars: the pattern `f x` is a HIGHER-ORDER unification problem: f x =?= s/2 — HOU: f := fun s => s/2, x := s is one solution; f := fun _ => s/2, x := anything another... The elaborator picked f := HDiv.hDiv s (function fun y => y/2? no — `HDiv.hDiv s` = fun y => s / y?? HDiv.hDiv s : ℂ → ℂ = fun y => HDiv.hDiv s y = s/y — WRONG branch of HOU) and x := 2: f x = s/2 ✓ that's the other solution! Classic HOU ambiguity — the elaborator chose f := HDiv.hDiv s, x := 2.\n\nFix: pin f explicitly: `AnalyticAt.comp (f := fun s : ℂ => s / 2) (g := Complex.digamma) (analyticAt_digamma h) h1` — wait the named args: AnalyticAt.comp (hg) (hf) with {f g x} implicit: provide (f := fun s : ℂ => s / 2): then hg : AnalyticAt ℂ digamma ((fun s => s/2) x) — with x to unify: hg := analyticAt_digamma h : AnalyticAt ℂ digamma (s/2): (fun s => s/2) x =?= s/2: x := s ✓ (beta HOU with f pinned is first-order ✓). Then hf := h1 : AnalyticAt ℂ (fun s => s/2) s ✓.\n\nSo: `have h2 : AnalyticAt ℂ (Complex.digamma ∘ fun s : ℂ => s / 2) s := AnalyticAt.comp (f := fun s : ℂ => s / 2) (analyticAt_digamma h) h1`.","signature":""},{"type":"text","text":"Classic higher-order unification ambiguity in `AnalyticAt.comp` — the elaborator picked `f := HDiv.hDiv s, x := 2`. Pinning `f` explicitly."},{"type":"tool_use","id":"Edit_130","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean","old_string":" have h2 : AnalyticAt ℂ (Complex.digamma ∘ fun s : ℂ => s / 2) s :=\n (analyticAt_digamma h).comp h1","new_string":" have h2 : AnalyticAt ℂ (Complex.digamma ∘ fun s : ℂ => s / 2) s :=\n AnalyticAt.comp (f := fun s : ℂ => s / 2) (analyticAt_digamma h) h1"},"extra_content":null}]