[{"type":"thinking","thinking":"`rw [seriesFun]` failed: \"Failed to rewrite using equation theorems for `seriesFun`\". Odd — seriesFun_zero at line 256 uses `rw [seriesFun]`... wait, that was in LiSeries.lean itself (same file where seriesFun is defined) — actually no, line 255-256 of LiMoments: `lemma seriesFun_zero : seriesFun 0 = 0 := by rw [seriesFun]` — that's in THIS file (LiMoments.lean line 256) and it compiled. So rw [seriesFun] works elsewhere in this file.\n\nWhy does it fail here? Goal: `iteratedDeriv j seriesFun 0 = ∑' ρ, ...`. rw [seriesFun] should rewrite the occurrence of `seriesFun` (the constant applied... actually `seriesFun 0` is an application `seriesFun 0`; the equation lemma `seriesFun.eq_1` or definitional unfold: `rw [seriesFun]` uses the equation `seriesFun = fun s => ...`? For rw to work with a definition, it uses the equational lemmas `seriesFun.eq_def` — rewriting occurrences of `seriesFun` (the head constant). In seriesFun_zero the goal is `seriesFun 0 = 0` — head constant present ✓. Here goal has `iteratedDeriv j seriesFun 0` — `seriesFun` present as a subterm ✓... but maybe the issue: equation theorems for seriesFun rewrite `seriesFun ?s = ∑' ρ, seriesTerm ?s ρ` (a lambda?? no). \"Failed to rewrite using equation theorems\" — the generated equation for a def without pattern matching is `seriesFun = fun s => ∑' ...`? Hmm, actually for simple defs Lean generates `seriesFun.eq_1 : seriesFun = fun s => ...`? Or `eq_def`. When I did it at line 256 it worked...\n\nWait — actually at line 256 the rw happens in a DIFFERENT context: `seriesFun 0 = 0` — after rw [seriesFun], goal `(∑' ρ, seriesTerm 0 ρ) = 0`? or `(fun s => ∑' ρ, seriesTerm s ρ) 0 = 0` which beta-reduces. It worked there.\n\nHere: the goal contains `iteratedDeriv j seriesFun 0`. Hmm, why \"failed to rewrite\"? Maybe because after the previous `obtain` and `have`s nothing changed the goal... The goal IS `iteratedDeriv j seriesFun 0 = ∑' ρ : subtype, ...`. Hmm — but wait: does the goal actually contain `seriesFun`? Yes: LHS.\n\nPossible subtlety: rw [seriesFun] uses equation lemmas tagged... for a plain def `def seriesFun (s : ℂ) : ℂ := ∑' ...`, the equational lemma is `seriesFun.eq_1 : ∀ (s : ℂ), seriesFun s = ∑' ρ, seriesTerm s ρ`? Hmm no — for non-recursive defs, Lean 4 generates `eq_def` only on demand? Actually `rw [seriesFun]` works by using the \"equation theorem\" `seriesFun.eq_1`/delta. The error \"Failed to rewrite using equation theorems\" specifically means Lean tried `seriesFun.eq_def`/`eq_1` and the rewrite failed to find the pattern OR the equational theorem doesn't exist.\n\nBut at line 256 in the SAME file it worked! So the pattern must be the issue: in `iteratedDeriv j seriesFun 0`, `seriesFun` appears unapplied (as a function argument). The equation theorem rewrites `seriesFun s` (applied form); `seriesFun` bare might not match the pattern `seriesFun ?s` via keyed matching — rw with `seriesFun` uses the equational lemma whose LHS is `seriesFun ✝` with a pattern variable; matching against bare `seriesFun` (no argument) fails... but eta: `seriesFun` = `fun s => seriesFun s`? Keyed matching wouldn't try eta-expansion. In seriesFun_zero, `seriesFun 0` IS applied ✓.\n\nFix: use `show` with the beta-expanded form, or `unfold seriesFun`, or eta-expand in the show: \n```lean\nshow iteratedDeriv j (fun s : ℂ => ∑' ρ : {ρ : ℂ // IsNontrivialZero ρ}, seriesTerm s ρ) 0 = _\n```\n`show` works up to defeq: `iteratedDeriv j seriesFun 0` vs `iteratedDeriv j (fun s => ∑' ρ, seriesTerm s ρ) 0` — needs seriesFun to unfold (default transparency defeq — `show` uses `isDefEq` with default transparency? `show` uses `withReducible`? No — `show` checks defeq at default transparency I believe; a plain def unfolds at default transparency? Regular defs unfold at default transparency, yes (only irreducible doesn't). ✓\n\nThen the next rw [← (iteratedDerivWithin_of_isOpen ...) h0mem] works on the lambda form. And iteratedDerivWithin_tsum's LHS pattern `(fun z => ∑' n, ?f n z)` matches syntactically.","signature":""},{"type":"text","text":"`rw [seriesFun]` fails on the unapplied constant (no eta-match). Using `show` with the unfolded form instead."},{"type":"tool_use","id":"Edit_58","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiMoments.lean","old_string":"  rw [seriesFun, ← (iteratedDerivWithin_of_isOpen Metric.isOpen_ball) h0mem,\n    iteratedDerivWithin_tsum j Metric.isOpen_ball h0mem\n      (fun t _ => seriesSummable t)\n      (fun k hk1 _ => seriesTerm_deriv_summableLocallyUniformlyOn hk1 hδ hδle)\n      ?_]","new_string":"  show iteratedDeriv j (fun s : ℂ => ∑' ρ : {ρ : ℂ // IsNontrivialZero ρ}, seriesTerm s ρ) 0 = _\n  rw [← (iteratedDerivWithin_of_isOpen Metric.isOpen_ball) h0mem,\n    iteratedDerivWithin_tsum j Metric.isOpen_ball h0mem\n      (fun t _ => seriesSummable t)\n      (fun k hk1 _ => seriesTerm_deriv_summableLocallyUniformlyOn hk1 hδ hδle)\n      ?_]"},"extra_content":null}]