[{"type":"thinking","thinking":"Wait — `collZeros_norm_le` uses `card96 t` but `card96` is defined AFTER it in my block. Need to move card96's def before collZeros_norm_le, or not use card96 there. Let me reorder: define card96 right after rC_gap (before collZeros). Actually in rC_gap the long expression `((scaledZeroFinset t (24/25) (by norm_num)).card : ℝ)` is what card96 abbreviates. Let me restructure: define card96 earlier and use it. But rC_gap's statement uses the explicit form... If I state rC_gap with card96 t, then collRadius_gap's explicit form must be converted: card96 t = that card by def (unfold). Simpler: keep rC_gap as-is, and in collZeros_norm_le don't use card96 — use the explicit form and add `unfold card96` issue away. Let me just inline the explicit form in collZeros_norm_le and define card96 after (for later use in Bbound etc.). Also then later lemmas can have a rewrite `card96_eq : (card96 t : ℝ) = ((scaledZeroFinset t (24/25) (by norm_num)).card : ℝ) := rfl`. Let me fix collZeros_norm_le to use the explicit form.","signature":""},{"type":"text","text":"`collZeros_norm_le` references `card96` before its definition — fixing by inlining."},{"type":"tool_use","id":"Edit_42","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDisc.lean","old_string":"/-- Collected zeros lie strictly inside: ‖w‖ ≤ r_c − δw. -/\ntheorem collZeros_norm_le {t : ℝ} (ht : 10 ≤ |t|) {w : ℂ} (hw : w ∈ collZeros ht) :\n    ‖w‖ ≤ rC ht - 1 / (50 * ((card96 t : ℝ) + 1)) := by\n  have hmem := (mem_collZeros ht).mp hw\n  have hw96 : w ∈ scaledZeroFinset t (24 / 25) (by norm_num) :=\n    scaledZeroFinset_subset (hr₂ := by norm_num) (rC_le ht) hw\n  have hgap := rC_gap ht hw96\n  have hle : ‖w‖ ≤ rC ht := hmem.1\n  have hX : ‖w‖ - rC ht ≤ 0 := sub_nonpos.mpr hle\n  rw [abs_of_nonpos hX] at hgap\n  unfold card96\n  linarith\n\n/-- The cardinality of the 24/25-finset. -/\nnoncomputable def card96 (t : ℝ) : ℕ :=\n  (scaledZeroFinset t (24 / 25) (by norm_num)).card\n\n/-- The mult-weighted count over the collected zeros, as a natural. -/\nnoncomputable def collMultSum {t : ℝ} (ht : 10 ≤ |t|) : ℕ :=\n  ∑ ρ ∈ collZeros ht, analyticOrderNatAt (scaledZeta t) ρ","new_string":"/-- Collected zeros lie strictly inside: ‖w‖ ≤ r_c − δw. -/\ntheorem collZeros_norm_le {t : ℝ} (ht : 10 ≤ |t|) {w : ℂ} (hw : w ∈ collZeros ht) :\n    ‖w‖ ≤ rC ht - 1 / (50 * (((scaledZeroFinset t (24 / 25)\n      (by norm_num : (24/25:ℝ) ≤ 1)).card : ℝ) + 1)) := by\n  have hmem := (mem_collZeros ht).mp hw\n  have hw96 : w ∈ scaledZeroFinset t (24 / 25) (by norm_num) :=\n    scaledZeroFinset_subset (hr₂ := by norm_num) (rC_le ht) hw\n  have hgap := rC_gap ht hw96\n  have hle : ‖w‖ ≤ rC ht := hmem.1\n  have hX : ‖w‖ - rC ht ≤ 0 := sub_nonpos.mpr hle\n  rw [abs_of_nonpos hX] at hgap\n  linarith\n\n/-- The cardinality of the 24/25-finset. -/\nnoncomputable def card96 (t : ℝ) : ℕ :=\n  (scaledZeroFinset t (24 / 25) (by norm_num)).card\n\n/-- The mult-weighted count over the collected zeros, as a natural. -/\nnoncomputable def collMultSum {t : ℝ} (ht : 10 ≤ |t|) : ℕ :=\n  ∑ ρ ∈ collZeros ht, analyticOrderNatAt (scaledZeta t) ρ"},"extra_content":null}]