[{"type":"thinking","thinking":"Progress: hC and hC2 now pass. New error at 988: `⊢ 0 ≤ ?m.2659` — a metavariable side goal. That's in `hfin`'s h1: `apply mul_le_mul_of_nonneg_left _ (by norm_num)` then `exact add_le_add_left (mul_le_mul_of_nonneg_left hΔle (by norm_num)) _`. The issue: `mul_le_mul_of_nonneg_left hΔle (by norm_num)` — the (by norm_num) arg is `0 ≤ ?c` where ?c is the multiplier — but the multiplier isn't determined yet (metavariable), so norm_num can't close `0 ≤ ?m.2659`. The multiplier is 0.532: `mul_le_mul_of_nonneg_left hΔle (show (0:ℝ) ≤ 0.532 by norm_num)`. Hmm — but even with `show 0 ≤ 0.532`, the elaborator must unify: mul_le_mul_of_nonneg_left (h : b ≤ a) (hc : 0 ≤ c) : c * b ≤ c * a. Target: 0.532·((y−16000)/16006.5) ≤ 0.532·((y−16000)/16000)?? wait the goal of h1's second part: 9.3564 + 0.532·((y−16000)/16006.5) ≤ 9.3564 + 0.532·((y−16000)/16000). add_le_add_left (h : b ≤ c) (a) : a + b ≤ a + c. So need b ≤ c: 0.532·Δ1 ≤ 0.532·Δ2 — mul_le_mul_of_nonneg_left hΔle (0 ≤ 0.532): but mul_le_mul_of_nonneg_left hΔle gives c·b ≤ c·a where hΔle : Δ1 ≤ Δ2: result c·Δ1 ≤ c·Δ2 ✓ with c := 0.532 — but the elaborator needs to figure c from hc's proof `by norm_num` which has goal `0 ≤ ?c` — metavariable. Fix: `mul_le_mul_of_nonneg_left hΔle (by norm_num : (0:ℝ) ≤ 0.532)` — wait, does that pin it? The ascription `(by norm_num : (0:ℝ) ≤ 0.532)` gives the proof term type 0 ≤ 0.532, so ?c := 0.532 ✓. \n\nActually — banked gotcha: ascribe numerals in such spots. Fix h1:\n\n```\nhave h1 : 416 * (9.3564 + 0.532 * ((y - 16000) / 16006.5))\n    ≤ 416 * (9.3564 + 0.532 * ((y - 16000) / 16000)) := by\n  apply mul_le_mul_of_nonneg_left _ (by norm_num : (0:ℝ) ≤ 416)\n  exact add_le_add_left\n    (mul_le_mul_of_nonneg_left hΔle (by norm_num : (0:ℝ) ≤ 0.532)) _\n```\n\nLet me apply.","signature":""},{"type":"text","text":"Metavariable in the un-ascribed multiplier. Pinning it."},{"type":"tool_use","id":"Edit_18","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"      apply mul_le_mul_of_nonneg_left _ (by norm_num)\n      exact add_le_add_left (mul_le_mul_of_nonneg_left hΔle (by norm_num)) _","new_string":"      apply mul_le_mul_of_nonneg_left _ (by norm_num : (0 : ℝ) ≤ 416)\n      exact add_le_add_left\n        (mul_le_mul_of_nonneg_left hΔle (by norm_num : (0 : ℝ) ≤ 0.532)) _"},"extra_content":null}]