[{"type":"thinking","thinking":"Line 847:50 is inside the `show (2 : ℝ) ≤ 2 * Real.sqrt (n : ℝ) by ...` block — the `rw [Real.le_sqrt (by norm_num) (by linarith)]; norm_num` at 848 — Real.le_sqrt's signature: `Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ Real.sqrt y ↔ x ^ 2 ≤ y` — hmm: `Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ √(y) ↔ x*x ≤ y`?? — the args: I passed `(by norm_num) (by linarith)` — the goals after rw should be `4 ^ 2 ≤ n` → norm_num/exact_mod_cast. The error at 847:50 (col 50) — the `⊢ 16 ≤ n` leftover suggests the hnR's exact_mod_cast failed inside the show block — wait, the context shows `hnR : 16 ≤ ↑n` PRESENT and `⊢ 16 ≤ n` as the goal — so the goal at the error is `16 ≤ n` (the ℕ proposition) — that's `Real.le_sqrt`'s SECOND argument `by linarith`?! — Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y)?? — maybe its signature is different: `Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ Real.sqrt y ↔ x ^ 2 ≤ y` — hmm — actually `Real.le_sqrt : 0 ≤ y → (x ≤ Real.sqrt y ↔ x ^ 2 ≤ y)`? or `Real.le_sqrt (hy : 0 ≤ y) (hx : 0 ≤ x)`... — the leftover goal `16 ≤ n` (ℕ!) suggests one of the args is a ℕ-side proof: Real.le_sqrt's args might include `0 ≤ (16:ℕ)`-ish... — let me check Real.le_sqrt's actual signature. Actually — simpler approach for `(4:ℝ) ≤ Real.sqrt (n:ℝ)` from (16:ℝ) ≤ n:\n```\n        have hsq : (4 : ℝ) ≤ Real.sqrt (n : ℝ) := by\n          have h1 : (4 : ℝ) ^ 2 ≤ (n : ℝ) := by\n            rw [show ((4 : ℝ) ^ 2) = (16 : ℝ) by norm_num]\n            exact_mod_cast hn\n          have h2 := Real.le_sqrt_of_sq_le (by positivity : (0:ℝ) ≤ (4:ℝ)) |>.mpr h1\n```\nHmm — `Real.le_sqrt_of_sq_le`? or `Real.le_sqrt (hx : 0 ≤ x) : x ≤ Real.sqrt y ↔ x ^ 2 ≤ y`... — simplest robust: use Real.sqrt_le_sqrt: `Real.sqrt 16 ≤ Real.sqrt n` from 16 ≤ n (cast) + `Real.sqrt 16 = 4`:\n```\n        have hsq : (4 : ℝ) ≤ Real.sqrt (n : ℝ) := by\n          have h1 : Real.sqrt 16 ≤ Real.sqrt (n : ℝ) :=\n            Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : ℝ) ≤ (n : ℝ))\n          have h2 : Real.sqrt 16 = 4 := by\n            rw [Real.sqrt_eq_iff_sq_eq (by norm_num) (by norm_num)]; norm_num\n          rwa [h2] at h1\n```\nReal.sqrt_le_sqrt (h : x ≤ y) : Real.sqrt x ≤ Real.sqrt y ✓ used before in the project ✓. Real.sqrt 16 = 4: Real.sqrt_eq_iff_sq_eq (hx : 0 ≤ x) (hy : 0 ≤ y) : √x = y ↔ y² = x — norm_num proves 4² = 16 ✓. This is robust. Replace the show-block.","signature":""},{"type":"text","text":"The `Real.le_sqrt` signature guess was wrong — using the robust `Real.sqrt_le_sqrt` route instead."},{"type":"tool_use","id":"Edit_359","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":"    have h2 := norm_logDeriv_zeta_le_two_pow\n      (show (2 : ℝ) ≤ 2 * Real.sqrt (n : ℝ) by\n        have hnR : (16 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn\n        have hsq : (4 : ℝ) ≤ Real.sqrt (n : ℝ) := by\n          rw [Real.le_sqrt (by norm_num) (by linarith)]; norm_num\n        linarith) (t := t)","new_string":"    have h2 := norm_logDeriv_zeta_le_two_pow\n      (show (2 : ℝ) ≤ 2 * Real.sqrt (n : ℝ) by\n        have hnR : (16 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn\n        have hsq : (4 : ℝ) ≤ Real.sqrt (n : ℝ) := by\n          have h1 : Real.sqrt 16 ≤ Real.sqrt (n : ℝ) :=\n            Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : ℝ) ≤ (n : ℝ))\n          have h2 : Real.sqrt 16 = 4 := by\n            rw [Real.sqrt_eq_iff_sq_eq (by norm_num) (by norm_num)]; norm_num\n          rwa [h2] at h1\n        linarith) (t := t)"},"extra_content":null}]