[{"type":"thinking","thinking":"The `positivity` tactic hits maxRecDepth on `0 < 2 * ((Z96.card : ℝ) + 1)` — probably because Z96 is a `set` variable and positivity tries to unfold it recursively. Fix: use explicit term proofs instead of positivity:\n\n0 < 2 * ((Z96.card : ℝ) + 1): (Z96.card : ℝ) ≥ 0 (Nat.cast_nonneg) ⟹ (k:ℝ)+1 ≥ 1 > 0 ⟹ 2*(k+1) > 0.\n\n```lean\n  have hpos2 : (0:ℝ) < 2 * ((Z96.card : ℝ) + 1) :=\n    mul_pos (by norm_num) (add_pos_of_nonneg_of_pos (Nat.cast_nonneg _) one_pos)\n```\nand similarly hpos:\n```lean\n  have hpos : (0:ℝ) < 20 * ((Z96.card : ℝ) + 1) :=\n    mul_pos (by norm_num) (add_pos_of_nonneg_of_pos (Nat.cast_nonneg _) one_pos)\n```\n\nAlso check the other positivity uses with the same pattern — in rC_gap I have hpos20/hpos50 with the full scaledZeroFinset expression — those might also loop? They compiled past line 547/552 only failing at these two — wait, only 2 errors reported, and rC_gap is LATER in the file — if earlier errors abort? No, Lean reports all errors. So rC_gap's positivities were fine?? Odd — maybe recursion came from Z96 being a `set`-bound local (positivity tries whnf of the let). rC_gap's expression uses scaledZeroFinset t (24/25) (by norm_num) directly — apparently fine. Let me just fix these two.","signature":""},{"type":"text","text":"`positivity` loops on the `set`-bound Z96. Replacing with explicit term proofs."},{"type":"tool_use","id":"Edit_45","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDisc.lean","old_string":"  have hpos2 : (0:ℝ) < 2 * ((Z96.card : ℝ) + 1) := by positivity\n  have h1 := (div_le_iff₀ hpos2).mp h\n  have h2 : 10 * |(5 / 2) * ‖w‖ - T / 10| * (2 * ((Z96.card : ℝ) + 1))\n      = |(5 / 2) * ‖w‖ - T / 10| * (20 * ((Z96.card : ℝ) + 1)) := by ring\n  rw [h2] at h1\n  have hpos : (0:ℝ) < 20 * ((Z96.card : ℝ) + 1) := by positivity\n  exact (div_le_iff₀ hpos).mpr h1","new_string":"  have hpos2 : (0:ℝ) < 2 * ((Z96.card : ℝ) + 1) :=\n    mul_pos (by norm_num) (add_pos_of_nonneg_of_pos (Nat.cast_nonneg _) one_pos)\n  have h1 := (div_le_iff₀ hpos2).mp h\n  have h2 : 10 * |(5 / 2) * ‖w‖ - T / 10| * (2 * ((Z96.card : ℝ) + 1))\n      = |(5 / 2) * ‖w‖ - T / 10| * (20 * ((Z96.card : ℝ) + 1)) := by ring\n  rw [h2] at h1\n  have hpos : (0:ℝ) < 20 * ((Z96.card : ℝ) + 1) :=\n    mul_pos (by norm_num) (add_pos_of_nonneg_of_pos (Nat.cast_nonneg _) one_pos)\n  exact (div_le_iff₀ hpos).mpr h1"},"extra_content":null}]