[{"type":"thinking","thinking":"Almost there. One error left: the `← mul_assoc` rewrite produced `2 * √↑n * √↑n` (which is `(2 * √n) * √n`), and my `rw [← mul_assoc 2 ...]` expected `2 * (√n * √n)` pattern — wrong direction. The goal is `↑n * 2 = 2 * √↑n * √↑n`. Just rewrite √n * √n directly: the RHS `2 * √n * √n` = `(2 * √n) * √n`. Use: `rw [← Real.mul_self_sqrt hnR.le]`... that rewrites `√n * √n` to n? Real.mul_self_sqrt : √x * √x = x — `rw [Real.mul_self_sqrt hnR.le]` would rewrite the pattern `√↑n * √↑n` — but the goal has `2 * √↑n * √↑n` parsed as `(2 * √↑n) * √↑n` — the pattern `√n * √n` doesn't appear syntactically (it's (2·√n)·√n). \n\nSimplest: `nlinarith [Real.mul_self_sqrt hnR.le]`? goal ↑n*2 = 2*√n*√n: from h : √n*√n = n: RHS = 2*(√n*√n) = 2n = LHS. As ring-normal: 2·√n·√n is the monomial 2·(√n·√n) — nlinarith should close via h: 2·√n·√n − 2n = 2(√n·√n − n) = 0 — it's linear in the atom (√n·√n)? nlinarith multiplies hypotheses: h × 2... linear combination: goal negated ↑n*2 ≠ 2*√n*√n... nlinarith handles equalities by splitting. Or use linear_combination: goal ↑n*2 = 2*√n*√n; want to subtract 2*(h.lhs − h.rhs): linear_combination 2 * Real.mul_self_sqrt hnR.le? The check: (↑n*2 − 2*√n*√n) − 2*(√n*√n − ↑n) = 2n − 2√n√n − 2√n√n + 2n = 4n − 4√n√n ≠ 0 ✗. Hmm: linear_combination uses coefficient: LHS−RHS = c·(e.lhs − e.rhs): need ↑n*2 − 2*√n*√n = c*(√n*√n − n): LHS−RHS = 2n − 2√n√n = −2(√n√n − n) ✓ c = −2: `linear_combination (-2) * Real.mul_self_sqrt hnR.le`? syntax: linear_combination -2 * Real.mul_self_sqrt hnR.le. Hmm — or simplest: replace the whole rw chain with:\n\n```lean\n          have h := Real.mul_self_sqrt hnR.le\n          nlinarith [h]\n```\ngoal ↑n * 2 = 2 * √↑n * √↑n: nlinarith splits into two ≤: 2n ≤ 2√n√n and reverse — from h (linear in atom √n√n once expanded): nlinarith normalizes products: 2*√n*√n becomes 2*(√n*√n)?? It ring-normalizes to monomials: √n*√n is a degree-2 monomial; h relates it to n. nlinarith considers h and products — should close directly (it's linear after treating √n*√n as an atom — nlinarith's preprocessing ring-nfs the goal and h to matching atoms). I'll try nlinarith [Real.mul_self_sqrt hnR.le].","signature":""},{"type":"text","text":"One rewrite direction issue left — the goal is `↑n * 2 = 2 * √↑n * √↑n`. Simplifying with nlinarith over the `√n*√n = n` fact."},{"type":"tool_use","id":"Edit_99","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean","old_string":"          rw [mul_one_div, div_eq_div_iff (by positivity : (2 * Real.sqrt (n : ℝ)) ≠ 0)\n            (by norm_num : (2 : ℝ) ≠ 0)]\n          rw [mul_comm (Real.sqrt (n : ℝ)) (2 * Real.sqrt (n : ℝ)),\n            ← mul_assoc 2 (Real.sqrt (n : ℝ)) (Real.sqrt (n : ℝ)),\n            Real.mul_self_sqrt hnR.le]\n          ring","new_string":"          rw [mul_one_div, div_eq_div_iff (by positivity : (2 * Real.sqrt (n : ℝ)) ≠ 0)\n            (by norm_num : (2 : ℝ) ≠ 0)]\n          nlinarith [Real.mul_self_sqrt hnR.le]"},"extra_content":null}]