[{"type":"thinking","thinking":"Two issues:\n\n1. **Line 738**: the `rw [div_eq_mul_inv]` — the goal's RHS is `(fun m => ↑k! / (↑m+1)^(k+1)) m` — an unreduced lambda application — div_eq_mul_inv's pattern ?a/?b not found because the RHS is `(fun m => ...) m` unbeta'd. Fix: beta-reduce first with `simp only []`? or `show`: \n```lean\n  rw [show (fun m => (k ! : ℝ) / ((m : ℝ) + 1) ^ (k + 1)) m = (k ! : ℝ) / ((m : ℝ) + 1) ^ (k + 1) from rfl]\n  rw [div_eq_mul_inv]\n```\nHmm — the `(fun m => ...) m` beta-reduction is definitional — a `show` with the beta'd form should work: `show ↑k ! * (‖s + ↑m + 1‖ ^ (k + 1))⁻¹ ≤ ↑k ! / (↑m + 1) ^ (k + 1)` then proceed. Actually the cleanest: use `refine le_trans ?_ (le_of_eq ?_)`... no. Just: `show (↑k ! : ℝ) * (‖s + ↑m + 1‖ ^ (k + 1))⁻¹ ≤ (k ! : ℝ) / ((m : ℝ) + 1) ^ (k + 1)` — the beta is defeq ✓ then continue with div_eq_mul_inv etc. But wait — after the show, I do rw [div_eq_mul_inv] on the target's RHS — and the one_div rewrites — let me redo the whole tail:\n```lean\n  show (↑k ! : ℝ) * (‖s + ↑m + 1‖ ^ (k + 1))⁻¹ ≤ (k ! : ℝ) / ((m : ℝ) + 1) ^ (k + 1)\n  rw [div_eq_mul_inv]\n  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n  rw [← one_div, ← one_div]  -- hmm — the goal: (↑k!)·(‖..‖^(k+1))⁻¹ ≤ (↑k!)·((↑m+1)^(k+1))⁻¹ — the ⁻¹ forms — one_div : 1/a = a⁻¹ — ← one_div turns a⁻¹ into 1/a ✓\n  apply one_div_le_one_div_of_le (by positivity)\n  exact pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (m : ℝ) + 1) hnormS (k + 1)\n```\nWait — after div_eq_mul_inv the goal's RHS: ↑k! * ((↑m+1)^(k+1))⁻¹ — and LHS ↑k! * (‖..‖^(k+1))⁻¹ — mul_le_mul_of_nonneg_left (h : (‖..‖^(k+1))⁻¹ ≤ ((↑m+1)^(k+1))⁻¹) (0 ≤ ↑k!) ✓ — the `apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)` leaves the ⁻¹-inequality — then rw [← one_div, ← one_div] turns both ⁻¹ into 1/· ✓ then one_div_le_one_div_of_le (0 < (m+1)^{k+1}) (h : (m+1)^{k+1} ≤ ‖s+..‖^{k+1})... — one_div_le_one_div_of_le (ha : 0 < a) (h : a ≤ b) : 1/b ≤ 1/a — so with a := (↑m+1)^{k+1}, b := ‖s+..‖^{k+1}: gives 1/b ≤ 1/a ✓ the goal after the one_div rewrites: 1/(‖..‖^{k+1}) ≤ 1/((↑m+1)^{k+1}) — hmm — that's 1/b ≤ 1/a form ✓ matches ✓.\n\nHmm wait — but the goal after `show` — the `show` changes the goal's RHS from `(fun m => ...) m` to the beta'd form — defeq ✓.\n\n2. **Line 768**: hf2's j ≥ 1 case: `rw [iteratedDerivWithin_dpsiTermC j (by omega) n hr]` — the goal is `DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r` — the rewrite pattern `iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r` — the goal has `(fun z => dpsiTermC z n)` with binder z vs the lemma's `(fun s => dpsiTermC s n)` — binder name shouldn't matter for rw (alpha-equiv is fine in Lean 4)... hmm — but the error says pattern not found — the issue: the lemma `iteratedDerivWithin_dpsiTermC j (by omega) n hr` — its conclusion's LHS is `iteratedDerivWithin j (fun s : ℂ => dpsiTermC s n) dpsiDomain s` with s := the lemma's {s} implicit — the application `... n hr` — hr : r ∈ dpsiDomain — so the lemma's s := r ✓ — conclusion's LHS: iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r — and the goal's function: (fun z => dpsiTermC z n) — same up to binder name — rw should match (alpha-equivalence is handled by the matcher? — rw matches up to syntactic equality modulo alpha ✓ usually fine)... Hmm the error says not found — maybe the issue is that the lemma's {s : ℂ} implicit and my application order: `iteratedDerivWithin_dpsiTermC j (by omega) n hr` — the signature: (k : ℕ) (hk : 1 ≤ k) (m : ℕ) {s : ℂ} (hs : s ∈ dpsiDomain) — so positional: k := j, hk := (by omega), m := n, then hs := hr — but s is IMPLICIT and comes between m and hs — the application `... j (by omega) n hr` — the implicit s should be inferred from hr : r ∈ dpsiDomain ✓ — conclusion: iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r = ... — and the goal: DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r — rw [that-lemma] at the goal — the pattern iteratedDerivWithin j (fun s => ...) dpsiDomain r — the goal has iteratedDerivWithin j (fun z => ...) dpsiDomain r — alpha-equiv... — hmm Lean 4 rw does handle binder-name alpha differences? — actually NO — rw matching is syntactic up to instances/reducible-defeq — alpha-equivalence (binder names) IS syntactic equality in Lean 4 (bound variables are de Bruijn) — binder NAMES are not part of the term — so (fun z => ...) and (fun s => ...) are THE SAME TERM (binder names are just display) — so they DO match... then why not found? — OH — maybe the issue: the goal's `iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain` — as an argument to DifferentiableAt — the pattern in the lemma conclusion is the same term — should match... — UNLESS the `by omega` for 1 ≤ j: the hk slot — `by omega` with j > 0 (hj0 : j > 0 in context) — hmm — omega should prove 1 ≤ j from j > 0 ✓...\n\nWait — actually look at the error context: `hj0 : j > 0` — and the goal — the error says pattern not found. Hmm — let me look at the goal: `DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r` — and the lemma conclusion LHS: `iteratedDerivWithin j (fun s : ℂ => dpsiTermC s m) dpsiDomain s` — with m := n, s := r — ✓ should match...\n\nHmm — unless the (by omega) failed and the whole application is malformed... — the error is at the rw — \"did not find the pattern iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r\" — hmm — actually WAIT — maybe the issue: `iteratedDerivWithin_dpsiTermC j (by omega) n hr` — the `by omega` targets hk : 1 ≤ j — the omega context has hj0 : j > 0 ✓ fine — and the resulting term's type: iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r = −(−1)^j·... — then rw [that] — the pattern is the LHS — the goal contains it — should fire. The only subtlety: the goal's lambda is `(fun z => dpsiTermC z n)` — binder z — and the lemma's is `(fun s => dpsiTermC s n)` — same de Bruijn term ✓...\n\nHmm — let me look at the error more carefully: \"Did not find an occurrence of the pattern iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r in the target expression DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r\" — these are the SAME up to binder display... UNLESS the m vs n: the lemma application `iteratedDerivWithin_dpsiTermC j (by omega) n hr` — m := n ✓... \n\nOH WAIT. I see it — the `hr` I pass: `hr : r ∈ dpsiDomain` — but the lemma's hypothesis is `hs : s ∈ dpsiDomain` with s implicit — the application `iteratedDerivWithin_dpsiTermC j (by omega) n hr` — the implicit {s : ℂ} — Lean infers s from hr's type r ∈ dpsiDomain → s := r ✓ — the conclusion: iteratedDerivWithin j ... dpsiDomain r ✓ — should match the goal...\n\nHmm — actually — maybe the issue is the `(by omega)` : the hk slot — with hj0 : j > 0 — omega proves 1 ≤ j ✓ — but is the resulting proof-term's TYPE exactly `1 ≤ j`? The lemma's hk : 1 ≤ k with k := j ✓.\n\nYou know what — let me look at the actual line 768 content — maybe there's a typo in MY code (like I wrote the lemma name slightly differently or the pattern in the goal is `iteratedDerivWithin j (fun z => dpsiTermC z n)` where dpsiTermC got unfolded...). From the error context: the goal is `DifferentiableAt ℂ (iteratedDerivWithin j (fun z => dpsiTermC z n) dpsiDomain) r` — the pattern in the rewrite: `iteratedDerivWithin j (fun s => dpsiTermC s n) dpsiDomain r` — same modulo binder — HOLD ON — `fun z => dpsiTermC z n` vs `fun s => dpsiTermC s n` — these are alpha-equivalent — Lean's rw SHOULD match... — UNLESS the issue is that rw [iteratedDerivWithin_dpsiTermC ...] with the lemma's conclusion being an equation — rw uses the equation's LHS as the pattern — the LHS is `iteratedDerivWithin k (fun s : ℂ => dpsiTermC s m) dpsiDomain s` with k, m, s being the lemma's VARIABLES — after application they're j, n, r ✓.\n\nUgh — let me just look at the actual current text around 760-770.","signature":""},{"type":"text","text":"Let me look at the actual text around the hf2 error."},{"type":"tool_use","id":"Read_431","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","offset":745,"limit":32},"extra_content":null}]