[{"type":"thinking","thinking":"The neg_neg rw failed because the term is `-(-(-1) ^ k * ...)` — note `-(-1) ^ k` parses as `-((-1)^k)` (the LiDigamma form). So the expression is `-( -((-1)^k) * k! * zpow )`. The pattern `- -?a` doesn't match because the outer structure is `-(X)` where X = `-((-1)^k) * k! * zpow` — wait, `-(-(-1) ^ k * ↑k ! * ...)`: the outer `-` applied to `(-(-1) ^ k * ↑k ! * zpow)`. Pattern `- -?a` requires the argument of outer neg to itself be a neg application `-?a`. Is `(-(-1) ^ k * ...)` a neg application? It's `HMul.hMul (-((-1)^k)) (...)`. The first factor is a neg, but the whole product isn't. So `- -?a` doesn't match.\n\nFix: the goal is `-((-((-1)^k)) * k! * zpow) = (-1)^k * k! * zpow`. Use `neg_mul`? Actually simplest: `simp only [neg_neg]`? neg_neg as simp would rewrite nested negs anywhere: `-((-((-1)^k)) * Y)` — simp only [neg_neg] rewrites `- -?a` patterns: the inner `-((-1)^k)` is `-(-(1^k))`... hmm `(-1)^k` — is that `Neg.neg (1^k)`? `(-1 : ℂ)` is `Neg.neg 1`. `(-1)^k` = `HPow.hPow (Neg.neg 1) k`. `-((-1)^k)` = `Neg.neg (HPow...)`. simp only [neg_neg] would turn `-(-1)^k`... wait the term `-(-1) ^ k` — how did it parse? In the goal display: `-(-1) ^ k * ↑k ! * (s + c) ^ (-↑k - 1)` with outer neg: `-(-(-1) ^ k * ...)`. Hmm the inner: `-(-1) ^ k` = `-((-1)^k)` per precedence. So inner term: `Neg.neg (HPow.hPow (Neg.neg 1) k)`. neg_neg pattern `- -?a`: `Neg.neg (Neg.neg ?a)` — is `HPow.hPow (Neg.neg 1) k` of the form `Neg.neg ?a`? No.\n\nSo the whole inner is not a double neg. The double neg is only at the OUTER level combined with... wait: goal LHS = `Neg.neg (HMul (Neg.neg ((-1)^k)) rest)`. Hmm no: `-(-(-1) ^ k * ↑k ! * zpow)` = Neg.neg(HMul(HMul(Neg.neg((-1)^k), k!), zpow)). The outer Neg.neg's argument is the product — not a neg. So neg_neg can't fire. \n\nI need: `-( (−((−1)^k)) * k! * zpow ) = (−1)^k * k! * zpow`. Note `(−((−1)^k)) * Y` = `−((−1)^k * Y)` via neg_mul (neg_mul : -a * b = -(a * b)). Then outer: `-(-((−1)^k * Y))` → neg_neg → `(−1)^k * Y`. So: rw [neg_mul, neg_neg]. neg_mul : -a * b = -(a * b) — the pattern `-?a * ?b`: the inner `(-(-1)^k) * (↑k! * zpow)`? The expression `-(-1) ^ k * ↑k ! * zpow` = `((-((-1)^k)) * ↑k !) * zpow`. rw [neg_mul] rewrites first match: `(-((-1)^k)) * ↑k!` → `-(((-1)^k) * ↑k!)`. Then expression: `(-(((-1)^k) * ↑k!)) * zpow`. rw [neg_mul] again → `-(((-1)^k * ↑k!) * zpow)`. Then outer neg: `-(-(((−1)^k * k!) * zpow))` → neg_neg → done.\n\nSo: rw [neg_mul, neg_mul, neg_neg]. Or simp only [neg_mul, neg_neg]. Let me use simp only [neg_mul, neg_neg] — rewrites repeatedly until normal form: LHS → (−1)^k * k! * zpow. Goal becomes X = X closed by rfl? simp closes. But careful simp only might also rewrite RHS? RHS has no neg patterns. Use `simp only [neg_mul, neg_neg]`.\n\nActually cleaner alternative: avoid the issue — after `rw [e, iteratedDeriv_neg k _ s, iteratedDeriv_neg_zpow_neg_one_add k c hs]`, replace final `neg_neg` with `simp only [neg_mul, neg_neg]`.","signature":""},{"type":"text","text":"The negative sign is buried inside the product. Fixing with `neg_mul` normalization."},{"type":"tool_use","id":"Edit_34","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiMoments.lean","old_string":"  rw [e, iteratedDeriv_neg k _ s, iteratedDeriv_neg_zpow_neg_one_add k c hs, neg_neg]","new_string":"  rw [e, iteratedDeriv_neg k _ s, iteratedDeriv_neg_zpow_neg_one_add k c hs]\n  simp only [neg_mul, neg_neg]"},"extra_content":null}]