[{"type":"thinking","thinking":"`add_le_add_left` in this Mathlib has the argument order swapped vs classic (banked gotcha from session 025: \"this Mathlib's add_le_add_left/right are swapped vs classic\"). `add_le_add_left (h : b ≤ c) (a)` might be `add_le_add_left (h : a ≤ b) : c + a ≤ c + b` as a CONSTRUCTOR without explicit a — the error shows it produced `x + ?m ≤ y + ?m` meaning the signature is `add_le_add_left : a ≤ b → ∀ (c), c + a ≤ c + b`?? The result type shows `0.532*... + ?m.2676 ≤ 0.532*... + ?m.2676` — so it's a + c ≤ b + c form — i.e., this Mathlib's `add_le_add_left (h : a ≤ b) (c) : a + c ≤ b + c`?? Actually classic: add_le_add_left (h : a ≤ b) (c) : c + a ≤ c + b. The error shows the c on the RIGHT: `x + ?m ≤ y + ?m`. So in this Mathlib, `add_le_add_left h c : a + c ≤ b + c`?? Hmm — the session-025 gotcha said they're swapped. Use `add_le_add_right` instead, or just use `by linarith` / `by gcongr`. Simplest robust: replace with `by gcongr` or provide the inequality via `add_le_add (le_refl _) (...)`. Let me just use:\n\n```\nexact add_le_add_left (mul_le_mul_of_nonneg_left hΔle (by norm_num : (0 : ℝ) ≤ 0.532)) (9.3564)\n```\nWait — if add_le_add_left h c : a + c ≤ b + c in this Mathlib (swapped), then add_le_add_left h 9.3564 gives 0.532Δ1 + 9.3564 ≤ 0.532Δ2 + 9.3564 — the 9.3564 is on the RIGHT of the sum, but our goal has it on the LEFT: 9.3564 + 0.532Δ1. `add` is commutative but not syntactically. So need add_le_add_right... if swapped, add_le_add_right h c : c + a ≤ c + b. Ugh. Safest: use gcongr:\n\n```\n· gcongr\n  exact mul_le_mul_of_nonneg_left hΔle (by norm_num : (0 : ℝ) ≤ 0.532)\n```\nHmm gcongr on 416·(9.3564+0.532·Δ1) ≤ 416·(9.3564+0.532·Δ2): gcongr should decompose: 416·_ monotone (needs 0 ≤ 416 side), then 9.3564+_ , then 0.532·_ (0 ≤ 0.532 side). gcongr leaves side goals: 0 ≤ 416? gcongr's pattern for mul_le_mul_of_nonneg_left requires positivity of the factor — it discharges via positivity usually. Let me just restructure h1 to avoid the apply:\n\n```\nhave h1 : 416 * (9.3564 + 0.532 * ((y - 16000) / 16006.5))\n    ≤ 416 * (9.3564 + 0.532 * ((y - 16000) / 16000)) :=\n  mul_le_mul_of_nonneg_left\n    (add_le_add (le_refl _)\n      (mul_le_mul_of_nonneg_left hΔle (by norm_num : (0 : ℝ) ≤ 0.532)))\n    (by norm_num : (0 : ℝ) ≤ 416)\n```\nadd_le_add (h1 : a ≤ b) (h2 : c ≤ d) : a + c ≤ b + d — unambiguous. a = 9.3564, b = 9.3564 (le_refl), c = 0.532·Δ1, d = 0.532·Δ2 ✓ gives 9.3564 + 0.532Δ1 ≤ 9.3564 + 0.532Δ2 ✓.\n\nAlso check the next error at 1037 — let me see what that is after fixing. Let me look at line 1037 region. It's probably in not_isNontrivialZero_on_gapCircle. Let me fix h1 first then recompile to see.","signature":""},{"type":"text","text":"This Mathlib's `add_le_add_left` is the swapped form (banked gotcha). Using `add_le_add` instead."},{"type":"tool_use","id":"Edit_20","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 : (0 : ℝ) ≤ 416)\n      exact add_le_add_left\n        (mul_le_mul_of_nonneg_left hΔle (by norm_num : (0 : ℝ) ≤ 0.532)) _","new_string":"      exact mul_le_mul_of_nonneg_left\n        (add_le_add (le_refl _)\n          (mul_le_mul_of_nonneg_left hΔle (by norm_num : (0 : ℝ) ≤ 0.532)))\n        (by norm_num : (0 : ℝ) ≤ 416)"},"extra_content":null}]