[{"type":"thinking","thinking":"Several fixes:\n\n1. **Line 43 (hLiExt_tendsto, hd)**: `h1.comp 0 h2` — the comp's second function g is inferred from h2 as `fun s => 1 + s`, but h1 is at point `1` and comp needs g 0 = 1 + 0 — the elaborator inferred g := Complex.mk 1?? The error: h2 expected to have type HasDerivAt (Complex.mk 1) ? 0 — the comp is trying to unify h1's point (1) with (g 0) where g := Complex.mk 1 (constant?!) — because `HasDerivAt.comp` has signature `HasDerivAt.comp (x : F) (hf : HasDerivAt f f' (h x)) (hh : HasDerivAt h h' x) : HasDerivAt (f ∘ h) x ...` — the (h x) in hf's type — with h1 : HasDerivAt hLi (deriv hLi 1) 1 — unifying (h x) with 1: h x := 1 — the elaborator picked h := fun _ => 1?? Because `1 + s` at `s := 0` — `1 + 0` needs to reduce to `1` — defeq but the elaborator didn't see through `1 + 0` when matching h1's point `1` against `(?h 0)` — it solved ?h := fun _ => 1 (constant) or Complex.mk 1 — hmm. In LiPrincipal's HIntegral_deriv, the same pattern worked: h1 : HasDerivAt G (deriv G (x + y*I)) (x + y*I); h2 : HasDerivAt (fun x => x + y*I) 1 x; hcomp := h1.comp x h2 — there h1's point was written AS (x + y*I) — the SAME expression as the path — so unification found ?G's point correctly. For mine: h1's point is the LITERAL 1, while g 0 = 1 + 0 — not syntactically 1. Fix: make h1's point `1 + 0` syntactically:\n```lean\n    have h1 : HasDerivAt hLi (deriv hLi 1) (1 + (0 : ℂ)) := by\n      simpa using hLi_analyticAt_one.differentiableAt.hasDerivAt\n```\nhmm — simpa using ... — the term's type HasDerivAt hLi (deriv hLi 1) 1 — ascribed to point (1 + 0) — defeq ✓ simpa/exact works. But then h1.comp 0 h2 with h2 : HasDerivAt (fun s => 1 + s) 1 0 — the comp: hf at (g 0) — g 0 = 1 + 0 ✓ syntactic match with h1's point (1 + 0)?? — (fun s => 1 + s) 0 — beta-reduces to 1 + 0 ✓ — the elaborator should unify ?point... Actually the issue might be deeper — HasDerivAt.comp for ℂ: `HasDerivAt.comp (x : F) (hf : HasDerivAt f f' (g x)) (hg : HasDerivAt g g' x) : HasDerivAt (f ∘ g) x (g' • f')`? — with the derivative `g' • f'` — for ℂ→ℂ: g' • f' = g' * f' (smul = mul). Hmm — but in LiPrincipal they got (deriv G p) * 1 via mul_one — so the result was (something)*1 — if the formula were g' • f' with g' = 1: 1 • f' — they'd need one_smul not mul_one — they used mul_one — so the comp result was f' * g' = f' * 1 — so the ℂ-to-ℂ comp gives f' * g'. OK.\n   Fix: h1 with point (1 + (0:ℂ)):\n```lean\n    have h1 : HasDerivAt hLi (deriv hLi 1) (1 + (0 : ℂ)) := by\n      have := hLi_analyticAt_one.differentiableAt.hasDerivAt\n      simpa using this\n```\nHmm — simpa using this : HasDerivAt hLi (deriv hLi 1) 1 — goal HasDerivAt hLi (deriv hLi 1) (1 + 0) — the point differs by 1+0 vs 1 — defeq — `simpa` normalizes the goal's 1+0 → 1? simp knows add_zero ✓ so simpa closes ✓. Then h1.comp 0 h2 — unifying h1's point (1 + 0) with (fun s => 1+s) 0 — beta: 1 + 0 ✓ match ✓.\n   Then h3 : HasDerivAt (hLi ∘ fun s => 1 + s) (deriv hLi 1 * 1) 0 — rw [mul_one] at h3 ✓ then the h4 rewrite — wait h4 : (hLi ∘ fun s => 1+s) = fun s => hLi (1+s) := rfl — rw [h4] at h3 — h3's function is hLi ∘ (fun s => 1+s) — h4's LHS is hLi ∘ fun s => 1+s — match ✓.\n   Hmm — but wait: with h1 at point (1+0), h3's result: HasDerivAt (hLi ∘ (fun s => 1+s)) (deriv hLi 1 * 1)? or (1 • deriv hLi 1)? — whichever, mul_one/one_smul — keep mul_one and adjust if the error says otherwise.\n\n2. **Line 88 (hs1 in hdiff)**: my rw [Metric.mem_ball] — the goal is `1 + s ∈ Metric.ball 1 ε` and I have hds : dist s 0 < ε. The rewrite failed because I tried `rw [Metric.mem_ball]` on... the error shows the goal `1 + s ∈ Metric.ball 1 ε` with hds : dist s 0 < ε — and my `rw [Metric.mem_ball]` at... the error says pattern `? ∈ Metric.ball` not found in target `dist s 0 < ε` — wait no: the error context shows the goal is `1 + s ∈ Metric.ball 1 ε` — the rw failure was on the PREVIOUS rw [Metric.mem_ball, dist_zero_right] at hds — hds was already `dist s 0 < ε`?? Looking at my code:\n```lean\n    have hs1 : (1 + s) ∈ Metric.ball 1 ε := by\n      have hds : dist s 0 < ε := lt_of_lt_of_le hs.1 (min_le_right _ _)\n      rw [Metric.mem_ball, dist_zero_right] at hds\n      ...\n```\nhs.1 : s ∈ Metric.ball 0 (min (δ/2) ε) — that's MEMBERSHIP not dist — lt_of_lt_of_le hs.1 fails?? hs.1 : s ∈ Metric.ball 0 _ — the lt_of_lt_of_le expects an inequality — TYPE MISMATCH would have been the error... but the reported error is at the rw line: \"pattern ? ∈ Metric.ball not found in dist s 0 < ε\" — hmm the error at 88:10 — line 88 is `rw [Metric.mem_ball]` (the one inside hs1's goal)... The error context shows the goal `1 + s ∈ Metric.ball 1 ε` — so `rw [Metric.mem_ball]` on the goal `1 + s ∈ Metric.ball 1 ε` — the pattern `?m ∈ Metric.ball ? ?` SHOULD match the goal... wait the error message says the pattern wasn't found in \"the target expression dist s 0 < ε\" — the target was dist s 0 < ε — that's from `rw [Metric.mem_ball, dist_zero_right] at hds` — but hds : dist s 0 < ε (after lt_of_lt_of_le gave... no wait — lt_of_lt_of_le hs.1 (min_le_right _ _) — hs.1 : s ∈ ball 0 (min..) — that's not an inequality! Unless Lean coerced... Actually hs : s ∈ Metric.ball 0 (min (δ/2) ε) \\ {0} — hs.1 : s ∈ Metric.ball 0 (min (δ/2) ε) — and `dist s 0 < ε := lt_of_lt_of_le hs.1 ...` — this should ERROR type mismatch — but the error shown is different... Hmm — actually looking at the context in the error: hds : dist s 0 < ε EXISTS — so the lt_of_lt_of_le DID typecheck — meaning hs.1 was elaborated as `dist s 0 < min (δ/2) ε`?? — because Metric.mem_ball is defeq to dist < — `s ∈ Metric.ball 0 r` unfolds to `dist s 0 < r` — so hs.1 : dist s 0 < min (δ/2) ε definitionally — lt_of_lt_of_le (that) (min_le_right ..) : dist s 0 < ε ✓ worked. THEN `rw [Metric.mem_ball, dist_zero_right] at hds` — hds : dist s 0 < ε — Metric.mem_ball pattern is a MEMBERSHIP — not in hds → ERROR ✓ that's the 88:10 error. Fix: drop that rw — hds is already the dist form. Then continue: goal 1 + s ∈ Metric.ball 1 ε: rw [Metric.mem_ball]: → dist (1+s) 1 < ε; show (1+s) − 1 = s by ring; rw — dist (1+s) 1 = ‖(1+s)−1‖ = ‖s‖ = dist s 0 — let me write:\n```lean\n    have hs1 : (1 + s) ∈ Metric.ball 1 ε := by\n      have hds : dist s 0 < ε := lt_of_lt_of_le hs.1 (min_le_right _ _)\n      rw [Metric.mem_ball]\n      have e : (1 + s) - 1 = s := by ring\n      rw [dist_eq_norm, e]\n      exact hds  -- hmm dist s 0 vs ‖s − 0‖: dist_eq_norm : dist a b = ‖a − b‖ — goal ‖(1+s)−1‖ < ε → after e: ‖s‖ < ε — hds : dist s 0 < ε — dist s 0 = ‖s − 0‖ = ‖s‖ — rw [dist_eq_norm] at hds? hds becomes ‖s − 0‖ < ε — sub_zero — need care:\n```\n```lean\n    have hs1 : (1 + s) ∈ Metric.ball 1 ε := by\n      have hds : dist s 0 < ε := lt_of_lt_of_le hs.1 (min_le_right _ _)\n      rw [Metric.mem_ball, dist_eq_norm]\n      have e : (1 + s) - 1 = s := by ring\n      rw [e]\n      rw [dist_zero_right] at hds\n      exact hds\n```\ndist_zero_right : dist s 0 = ‖s‖ ✓ so hds : ‖s‖ < ε ✓ and the goal after rewrites: ‖s‖ < ε ✓ exact hds ✓. Wait — dist_eq_norm : dist a b = ‖a − b‖ — applied at goal dist (1+s) 1 → ‖(1+s) − 1‖ ✓ then e ✓ → ‖s‖ < ε ✓ and hds after dist_zero_right: ‖s‖ < ε ✓.\n\n3. **Line 139/141 (hLiExt_mul_id)**: after funext s, the goal is `((fun s => s) * hLiExt) s = 1 − hLi (1 + s)` — the Pi.mul application not beta-reduced — rw [hLiExt_zero] failed because the goal has `((fun s => s) * hLiExt) 0` not `hLiExt 0` syntactically visible... it IS visible inside: ((fun s => s) * hLiExt) 0 — the pattern hLiExt 0 doesn't occur (the term is (⋯ * hLiExt) 0 — hLiExt appears but not applied to 0 alone). Fix: simp only [Pi.mul_apply] first to beta-reduce: goal → (fun s => s) 0 * hLiExt 0 = ... hmm — better: \n```lean\n  funext s\n  by_cases hs : s = 0\n  · subst hs\n    show (0 : ℂ) * hLiExt 0 = 1 - hLi (1 + 0)\n    rw [hLiExt_zero, hLi_one]\n    simp\n  · show s * hLiExt s = 1 - hLi (1 + s)\n    rw [hLiExt_of_ne hs]\n    ...\n```\nThe `show` uses defeq (Pi.mul_apply) ✓ — then hLiExt 0 appears ✓. And the second branch's show: s * hLiExt s = 1 − hLi (1+s) ✓ then rw [hLiExt_of_ne hs] : hLiExt s → (1 − hLi(1+s))/s ✓ — then the previous show line I had: `show s * ((1 − hLi (1 + s)) / s) = 1 - hLi (1 + s)` — redundant now; directly rw [mul_div_assoc', mul_div_cancel_left₀ _ hs] ✓.\n\n4. **Line 152 (iteratedDeriv_hLi_one_add)**: e : iteratedDeriv m (fun z => hLi (z − −1)) 0 = iteratedDeriv m hLi (0 − −1) — so iteratedDeriv_comp_sub_const ALREADY gives the lambda form (fun z => hLi (z − −1)) not the ∘ form! My e2 was about the ∘ form. Fix e2:\n```lean\n  have e := congr_fun (iteratedDeriv_comp_sub_const m hLi (-1 : ℂ)) 0\n  -- e : iteratedDeriv m (fun z => hLi (z - -1)) 0 = iteratedDeriv m hLi (0 - -1)\n  have e2 : (fun z : ℂ => hLi (z - -1)) = fun s : ℂ => hLi (1 + s) := by\n    funext z\n    congr 1\n    ring\n  rw [e2] at e\n  simpa using e\n```\nsimpa: 0 − −1 → 1 ✓ (sub_neg_eq_add, zero_add — simp ✓).\n\n5. **Line 193 (hid, i ≥ 2 case)**: the induction: \n```lean\n      have h2 : iteratedDeriv (i + 2) (fun s : ℂ => s) = fun _ => (0 : ℂ) := by\n        induction i with\n        | zero =>\n          funext s\n          rw [iteratedDeriv_succ', iteratedDeriv_one, deriv_id'']\n          simp\n        | succ i ih =>\n          rw [iteratedDeriv_succ', ih]\n          simp\n      rw [h2]\n      simp\n```\nThe error at 193:35: pattern `iteratedDeriv (i + 2) (fun s => s)` not found in `iteratedDeriv (i + 2) (deriv fun s => s) = fun x => 0` — that's the succ case: after rw [iteratedDeriv_succ']: goal became iteratedDeriv (i+2)... wait the context shows the succ case goal AFTER `rw [iteratedDeriv_succ', ih]` partially: the error is that rw [ih] failed? The message: pattern `iteratedDeriv (i + 2) (fun s => s)` not found in target `iteratedDeriv (i + 2) (deriv fun s => s) = fun x => 0` — hmm — this is the `succ i` case where the goal is iteratedDeriv (i+1+2) (fun s => s) = 0 — rw [iteratedDeriv_succ'] : iteratedDeriv (i+1+2) f = deriv (iteratedDeriv (i+1+1) f)?? — iteratedDeriv_succ' : iteratedDeriv (n+1) f = deriv (iteratedDeriv n f)? — so iteratedDeriv (i+3) (fun s => s) → deriv (iteratedDeriv (i+2) (fun s => s)) — wait but the error's target shows `iteratedDeriv (i + 2) (deriv fun s => s)` — so iteratedDeriv_succ' rewrote to iteratedDeriv (i+2) (deriv (fun s => s))!! — meaning iteratedDeriv_succ' : iteratedDeriv (n+1) f = iteratedDeriv n (deriv f)!! (the OTHER convention — deriv applied inside). So after rw [iteratedDeriv_succ']: goal iteratedDeriv (i+2) (deriv (fun s => s)) = 0 — and ih : iteratedDeriv (i+2) (fun s => s) = fun _ => 0 — the deriv-inside form doesn't match ih. Hmm — wait — actually let me recheck: the error occurred in the SUCC branch — the target `iteratedDeriv (i + 2) (deriv fun s => s) = fun x => 0` — and my rw [ih] looked for iteratedDeriv (i+2) (fun s => s) — not found ✓ that's the error. \n\nRework hid's i ≥ 2 case. Since iteratedDeriv_succ' : iteratedDeriv (n+1) f = iteratedDeriv n (deriv f)?? — hmm wait, let me double check what iteratedDeriv_succ' actually says — from iteratedDeriv_const's proof: `simp [iteratedDeriv_succ', h]` — for the const function. And A_iter/LiResidue used iteratedDeriv_succ'... The two lemmas: iteratedDeriv_succ : iteratedDeriv (n+1) f = deriv (iteratedDeriv n f); iteratedDeriv_succ' : iteratedDeriv (n+1) f = iteratedDeriv n (deriv f)? Hmm — from the error, iteratedDeriv_succ' gave iteratedDeriv (i+2) (deriv (fun s => s)) — so succ' moves deriv INSIDE ✓.\n\nRedo hid's i≥2 case using deriv-first:\n```lean\n    · -- i ≥ 2: iteratedDeriv (i+2) id = iteratedDeriv (i+1) (deriv id) = iteratedDeriv (i+1) (fun _ => 1) = 0\n      have h2 : iteratedDeriv (i + 2) (fun s : ℂ => s) = fun _ => (0 : ℂ) := by\n        funext s\n        rw [iteratedDeriv_succ']  -- iteratedDeriv (i+2) id s = iteratedDeriv (i+1) (deriv id) s\n        rw [deriv_id'']           -- deriv id → fun s => 1? deriv_id'' : deriv id = fun _ => 1? hmm deriv_id'' : deriv (fun x => x) x = 1 (pointwise!)\n        ...\n```\nderiv_id'' is POINTWISE: deriv (fun x => x) x = 1? or deriv id x = 1 — the form `deriv_id'' : deriv (fun x : 𝕜 => x) x = 1`? — hmm there are deriv_id (deriv id x = 1), deriv_id' (deriv (fun x => x) x = 1?), deriv_id'' — pointwise versions. The iteratedDeriv machinery's const lemma handles const functions: after getting iteratedDeriv (i+1) (fun _ => 1) — the deriv (fun s => s) as a FUNCTION is fun _ => 1 — is deriv (fun s => s) SYNTACTICALLY fun _ => 1? No — deriv f is an application. Use: iteratedDeriv (i+1) (deriv (fun s => ℂ => s)) s — rewrite deriv (fun s => s) to (fun _ => 1) — the function equality: `deriv_id''`? no — need function-level: have : deriv (fun s : ℂ => s) = fun _ => 1 := by funext s; exact deriv_id'' s? — deriv_id'' s : deriv (fun x => x) s = 1? — hmm deriv_id'' : deriv (fun x => x) x = 1 — I'm not sure of its exact statement (could be deriv id x = 1). funext + exact deriv_id''/deriv_id: \n```lean\n      have hd : deriv (fun s : ℂ => s) = fun _ => (1 : ℂ) := by\n        funext s\n        exact deriv_id'' s  -- or deriv_id' s / deriv_id s\n      rw [iteratedDeriv_succ', hd]\n      -- iteratedDeriv (i+1) (fun _ => 1) s = if i+1 = 0 then 1 else 0 = 0\n      rw [iteratedDeriv_const, if_neg (by omega)]\n```\nWait — but iteratedDeriv_const's conclusion is `if n = 0 then c else 0` with the VALUE — iteratedDeriv (i+1) (fun _ => 1) s = if i+1 = 0 then 1 else 0 ✓ if_neg (i+1 ≠ 0 by omega) → 0 ✓.\n\nHmm — but actually the `funext s` + pointwise: h2's goal: iteratedDeriv (i+2) (fun s => s) = fun _ => 0 — funext s: iteratedDeriv (i+2) (fun s => s) s = 0 — rw [iteratedDeriv_succ']: iteratedDeriv (i+2) f s = iteratedDeriv (i+1) (deriv f) s ✓ (succ' pointwise?) — iteratedDeriv_succ' is a function equality `iteratedDeriv (n+1) f = iteratedDeriv n (deriv f)`?? or pointwise `iteratedDeriv (n+1) f x = iteratedDeriv n (deriv f) x`? — from iteratedDeriv_const's proof `simp [iteratedDeriv_succ', h]` — ambiguous. If it's function-level: rw [iteratedDeriv_succ'] on the funext'd goal iteratedDeriv (i+2) (fun s => s) s = 0 — rewrites the iteratedDeriv (i+2) (fun s => s) subterm → iteratedDeriv (i+1) (deriv (fun s => s)) s ✓ fine either way.\n\nThen rw [hd] → iteratedDeriv (i+1) (fun _ => 1) s = 0 — rw [iteratedDeriv_const, if_neg ...] ✓.\n\nAlso the FIRST branch (i = 0): `simp [iteratedDeriv_zero]` — goal: iteratedDeriv 0 (fun s => s) 0 = if 0 = 1 then 1 else 0 — iteratedDeriv_zero : iteratedDeriv 0 f = f — simp: (fun s => s) 0 = 0 ✓ and if 0 = 1... — `if_neg`/simp ✓ should close. The second (i = 1): iteratedDeriv 1 (fun s => s) 0 = if 1 = 1 then 1 else 0 — simp [iteratedDeriv_one, deriv_id''] — iteratedDeriv_one : iteratedDeriv 1 f = deriv f; deriv_id''/deriv_id: deriv (fun s => s) 0 = 1 ✓ if_pos ✓ — simp should close — but is deriv_id'' the right simp lemma name? `deriv_id'' : deriv (fun x => x) x = 1`? There are deriv_id, deriv_id', deriv_id'' variants — simp [deriv_id''] was used in LiResidue (line 119: `simp [deriv_id'']` for deriv (fun z => z−1) — hmm that was deriv (fun z => z − 1) via deriv_id''? they had h2 : deriv (fun z => z - 1) w = 1 := by simp [deriv_id''] — so deriv_id'' is a simp lemma that helps with id-like functions ✓). For plain deriv (fun s => s) 0: simp [iteratedDeriv_one, deriv_id''] should close ✓.\n\nLet me also double check `iteratedDeriv_succ'`'s direction — from the error, at succ case: after `rw [iteratedDeriv_succ']` the goal's iteratedDeriv (i+1+2)... — the error target was `iteratedDeriv (i + 2) (deriv fun s => s) = fun x => 0` — hmm wait that's confusing: the succ case is `i+1`: goal iteratedDeriv (i+1+2) (fun s => s) = fun _ => 0 — after rw [iteratedDeriv_succ']: iteratedDeriv (i+1) (deriv (fun s=>s))?? — the error shows iteratedDeriv (i+2)(deriv ...) — so the succ case's i got displayed as i... whatever — the KEY fact: iteratedDeriv_succ' : iteratedDeriv (n+1) f = iteratedDeriv n (deriv f) ✓ confirmed by the error. Use it as such.\n\nSimplest restructure of hid avoiding induction:\n```lean\n  have hid : ∀ i : ℕ, iteratedDeriv i (fun s : ℂ => s) 0 = if i = 1 then 1 else 0 := by\n    intro i\n    rcases i with _ | _ | i\n    · simp [iteratedDeriv_zero]\n    · simp [iteratedDeriv_one, deriv_id'']\n    · have hd : deriv (fun s : ℂ => s) = fun _ => (1 : ℂ) := by\n        funext s\n        exact deriv_id'' s\n      have h2 : iteratedDeriv (i + 2) (fun s : ℂ => s) 0 = 0 := by\n        rw [iteratedDeriv_succ', hd, iteratedDeriv_const, if_neg (by omega)]\n      rw [h2]\n      simp\n```\nWait — the rcases third case: i becomes i+2 — the goal: iteratedDeriv (i+2) (fun s => s) 0 = if (i+2) = 1 then 1 else 0 — rw [h2] — h2 says LHS = 0 — then goal 0 = if i+2 = 1 then 1 else 0 — simp (if_neg, i+2 ≠ 1 by omega — simp's omega? `simp` might not close if i+2 = 1 evaluation — need if_neg (by omega : i + 2 ≠ 1)... — write: `rw [h2]; simp` — simp knows i+2 ≠ 1? via `Nat.succ_ne...`? i+2 = (i+1)+1 — simp might not close 0 = if i+2 = 1 then ... — the if condition i+2 = 1 — simp can evaluate? i is a variable — the condition is not decidable by simp alone... Use: `rw [h2, if_neg (by omega)]` then rfl — wait after rw [h2, if_neg ...] : goal 0 = 0 — rfl auto by rw ✓. Hmm if_neg needs the proof ¬(i+2 = 1): `by omega` ✓.\n\nSimilarly first two branches: rcases _ | _ | i gives i=0: if 0 = 1 then 1 else 0 — simp closes via if_neg? 0 ≠ 1 — decidable — simp evaluates the if (decide) ✓. i=1: if 1 = 1 — if_pos ✓ simp ✓.\n\nNow also fix hLiExt_mul_id with the show-beta bridge and hdiff's hs1. And check hcompC in iteratedDeriv_hLiExt: h1a : AnalyticAt ℂ hLi (1 + (0:ℂ)) := by simpa using hLi_analyticAt_one — hLi_analyticAt_one : AnalyticAt ℂ hLi 1 — goal AnalyticAt ℂ hLi (1+0) — simpa (add_zero → 1) ✓. h1a.comp h2a : AnalyticAt ℂ (hLi ∘ fun s => 1+s) 0 — AnalyticAt.comp signature: (hf : AnalyticAt 𝕜 f (g x)) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f ∘ g) x — h1a at point (1+0) and h2a : AnalyticAt (fun s => 1+s) 0 — need g 0 = (1+0) — (fun s => 1+s) 0 = 1+0 ✓ syntactic-ish — should unify ✓ (here the point IS written 1+0 ✓).\n\nWait — actually the same issue as hLiExt_tendsto could bite: h1a : AnalyticAt ℂ hLi (1 + 0) — simpa using hLi_analyticAt_one : AnalyticAt ℂ hLi 1 — goal (1+0): simpa normalizes goal 1+0 → 1 via add_zero? simp includes add_zero ✓ so simpa using ... closes AnalyticAt ℂ hLi (1+0) with the term of type ... hLi 1 — defeq/simp ✓.\n\nLet me apply all fixes.","signature":""},{"type":"text","text":"Applying fixes: comp point syntax, ball-membership rewrite order, Pi.mul beta, comp-const lambda form, hid restructure."},{"type":"tool_use","id":"Edit_196","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean","old_string":"  have hd : HasDerivAt (fun s : ℂ => hLi (1 + s)) (deriv hLi 1) 0 := by\n    have h1 : HasDerivAt hLi (deriv hLi 1) 1 := hLi_analyticAt_one.differentiableAt.hasDerivAt\n    have h2 : HasDerivAt (fun s : ℂ => 1 + s) 1 0 := by\n      simpa using (hasDerivAt_id (0 : ℂ)).const_add 1\n    have h3 := h1.comp 0 h2\n    rw [mul_one] at h3\n    have h4 : (hLi ∘ fun s : ℂ => 1 + s) = fun s : ℂ => hLi (1 + s) := rfl\n    rw [h4] at h3\n    exact h3","new_string":"  have hd : HasDerivAt (fun s : ℂ => hLi (1 + s)) (deriv hLi 1) 0 := by\n    have h1 : HasDerivAt hLi (deriv hLi 1) (1 + (0 : ℂ)) := by\n      simpa using hLi_analyticAt_one.differentiableAt.hasDerivAt\n    have h2 : HasDerivAt (fun s : ℂ => 1 + s) 1 0 := by\n      simpa using (hasDerivAt_id (0 : ℂ)).const_add 1\n    have h3 := h1.comp 0 h2\n    rw [mul_one] at h3\n    have h4 : (hLi ∘ fun s : ℂ => 1 + s) = fun s : ℂ => hLi (1 + s) := rfl\n    rw [h4] at h3\n    exact h3"},"extra_content":null},{"type":"tool_use","id":"Edit_197","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean","old_string":"    have hs1 : (1 + s) ∈ Metric.ball 1 ε := by\n      have hds : dist s 0 < ε := lt_of_lt_of_le hs.1 (min_le_right _ _)\n      rw [Metric.mem_ball, dist_zero_right] at hds\n      rw [Metric.mem_ball]\n      have : (1 + s) - 1 = s := by ring\n      rw [this]\n      exact hds","new_string":"    have hs1 : (1 + s) ∈ Metric.ball 1 ε := by\n      have hds : dist s 0 < ε := lt_of_lt_of_le hs.1 (min_le_right _ _)\n      rw [Metric.mem_ball, dist_eq_norm]\n      have e : (1 + s) - 1 = s := by ring\n      rw [e, ← dist_zero_right]\n      exact hds"},"extra_content":null},{"type":"tool_use","id":"Edit_198","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean","old_string":"  funext s\n  by_cases hs : s = 0\n  · subst hs\n    rw [hLiExt_zero, hLi_one]\n    simp\n  · rw [hLiExt_of_ne hs]\n    show s * ((1 - hLi (1 + s)) / s) = 1 - hLi (1 + s)\n    rw [mul_div_assoc', mul_div_cancel_left₀ _ hs]","new_string":"  funext s\n  by_cases hs : s = 0\n  · subst hs\n    show (0 : ℂ) * hLiExt 0 = 1 - hLi (1 + 0)\n    rw [hLiExt_zero, hLi_one]\n    simp\n  · show s * hLiExt s = 1 - hLi (1 + s)\n    rw [hLiExt_of_ne hs, mul_div_assoc', mul_div_cancel_left₀ _ hs]"},"extra_content":null},{"type":"tool_use","id":"Edit_199","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean","old_string":"  have e := congr_fun (iteratedDeriv_comp_sub_const m hLi (-1 : ℂ)) 0\n  have e2 : (hLi ∘ (· - (-1 : ℂ))) = (fun s : ℂ => hLi (1 + s)) := by\n    funext s\n    simp [Function.comp_apply, sub_neg_eq_add, add_comm]\n  rw [e2] at e\n  simpa using e","new_string":"  have e := congr_fun (iteratedDeriv_comp_sub_const m hLi (-1 : ℂ)) 0\n  have e2 : (fun z : ℂ => hLi (z - -1)) = fun s : ℂ => hLi (1 + s) := by\n    funext z\n    congr 1\n    ring\n  rw [e2] at e\n  simpa using e"},"extra_content":null},{"type":"tool_use","id":"Edit_200","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean","old_string":"  have hid : ∀ i : ℕ, iteratedDeriv i (fun s : ℂ => s) 0 = if i = 1 then 1 else 0 := by\n    intro i\n    rcases i with _ | _ | i\n    · simp [iteratedDeriv_zero]\n    · simp [iteratedDeriv_one, deriv_id'']\n    · have h2 : iteratedDeriv (i + 2) (fun s : ℂ => s) = fun _ => (0 : ℂ) := by\n        induction i with\n        | zero =>\n          funext s\n          rw [iteratedDeriv_succ', iteratedDeriv_one, deriv_id'']\n          simp\n        | succ i ih =>\n          rw [iteratedDeriv_succ', ih]\n          simp\n      rw [h2]\n      simp","new_string":"  have hid : ∀ i : ℕ, iteratedDeriv i (fun s : ℂ => s) 0 = if i = 1 then 1 else 0 := by\n    intro i\n    rcases i with _ | _ | i\n    · simp [iteratedDeriv_zero]\n    · simp [iteratedDeriv_one, deriv_id'']\n    · have hd : deriv (fun s : ℂ => s) = fun _ => (1 : ℂ) := by\n        funext s\n        exact deriv_id'' s\n      have h2 : iteratedDeriv (i + 2) (fun s : ℂ => s) 0 = 0 := by\n        rw [iteratedDeriv_succ', hd, iteratedDeriv_const, if_neg (by omega)]\n      rw [h2, if_neg (by omega)]"},"extra_content":null}]