← Session 31 — ii′-5f: tight R-bound 10.5/48 (piece 3 + pinch)
thinking (1426 chars)
Wait — I still have that broken h10 calc block (the `_ = (2*(1+t)^2)*(1+ρ.im²) − ... + ... := by ring` — the final step doesn't match the div_le_div_iff₀ goal `1 * (1+γ²)... ` hold on: div_le_div_iff₀ (hb : 0 < b) (hd : 0 < d) : a/b ≤ c/d ↔ a*d ≤ c*b — with a = 1, b = |x|², c = 2(1+t)², d = 1+γ²: goal: 1 * (1+γ²) ≤ (2(1+t)²) * |x|². My calc: (1:ℝ)*|x|² — that's a*d?? NO: a*d = 1 * (1+γ²) — my calc starts (1:ℝ)*|x|² = |x|² — WRONG target. The goal after div_le_div_iff₀ is `1 * (1 + ρ.1.im ^ 2) ≤ (2 * (1 + t) ^ 2) * |ρ.1.im - t| ^ 2` — and h4 says 1+γ² ≤ 2(1+t)²·x² — directly: `calc (1:ℝ) * (1+γ²) = 1+γ² := by ring; _ ≤ 2(1+t)²x² := h4; _ = (2(1+t)²)*x² := by ring`. Let me fix the h10 block:
```
have h10 : 1 / |ρ.1.im - t| ^ 2 ≤ (2 * (1 + t) ^ 2) / (1 + ρ.1.im ^ 2) := by
rw [div_le_div_iff₀ (by positivity : (0 : ℝ) < 1 + ρ.1.im ^ 2)
(by positivity : (0 : ℝ) < |ρ.1.im - t| ^ 2)]
calc (1 : ℝ) * (1 + ρ.1.im ^ 2) = 1 + ρ.1.im ^ 2 := by ring
_ ≤ 2 * (1 + t) ^ 2 * |ρ.1.im - t| ^ 2 := h4
_ = (2 * (1 + t) ^ 2) * |ρ.1.im - t| ^ 2 := by ring
```
— wait but which is c*b: c = 2(1+t)², b = |x|²: c*b = (2(1+t)²)·|x|² ✓ and a*d = 1·(1+γ²) ✓. The calc's last step: 2(1+t)²·|x|² = (2(1+t)²)·|x|² ✓ ring-equal (same thing) — actually the middle and last are syntactically identical (2*(1+t)^2*|x|^2 vs (2*(1+t)^2)*|x|^2) — ring can close or just drop the last step. Let me fix via Edit.
Fixing the broken h10 calc block before compiling.
tool: Edit
input
{
"replace_all": false,
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean",
"old_string": " have h10 : 1 / |\u03c1.1.im - t| ^ 2 \u2264 (2 * (1 + t) ^ 2) / (1 + \u03c1.1.im ^ 2) := by\n rw [div_le_div_iff\u2080 (by positivity : (0 : \u211d) < 1 + \u03c1.1.im ^ 2)\n (by positivity : (0 : \u211d) < |\u03c1.1.im - t| ^ 2)]\n calc (1 : \u211d) * |\u03c1.1.im - t| ^ 2 = |\u03c1.1.im - t| ^ 2 := by ring\n _ \u2264 1 + \u03c1.1.im ^ 2 := by\n nlinarith [sq_nonneg \u03c1.1.im]\n _ \u2264 2 * (1 + t) ^ 2 * |\u03c1.1.im - t| ^ 2 := h4\n _ = (2 * (1 + t) ^ 2) * (1 + \u03c1.1.im ^ 2) - (2 * (1 + t) ^ 2) * \u03c1.1.im ^ 2\n + ((2 * (1 + t) ^ 2) * \u03c1.1.im ^ 2 - (2 * (1 + t) ^ 2) * |\u03c1.1.im - t| ^ 2)\n + (2 * (1 + t) ^ 2) * |\u03c1.1.im - t| ^ 2 := by ring",
"new_string": " have h10 : 1 / |\u03c1.1.im - t| ^ 2 \u2264 (2 * (1 + t) ^ 2) / (1 + \u03c1.1.im ^ 2) := by\n rw [div_le_div_iff\u2080 (by positivity : (0 : \u211d) < 1 + \u03c1.1.im ^ 2)\n (by positivity : (0 : \u211d) < |\u03c1.1.im - t| ^ 2)]\n calc (1 : \u211d) * (1 + \u03c1.1.im ^ 2) = 1 + \u03c1.1.im ^ 2 := by ring\n _ \u2264 2 * (1 + t) ^ 2 * |\u03c1.1.im - t| ^ 2 := h4"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Wait — I still have that broken h10 calc block (the `_ = (2*(1+t)^2)*(1+ρ.im²) − ... + ... := by ring` — the final step doesn't match the div_le_div_iff₀ goal `1 * (1+γ²)... ` hold on: div_le_div_iff₀ (hb : 0 < b) (hd : 0 < d) : a/b ≤ c/d ↔ a*d ≤ c*b — with a = 1, b = |x|², c = 2(1+t)², d = 1+γ²: goal: 1 * (1+γ²) ≤ (2(1+t)²) * |x|². My calc: (1:ℝ)*|x|² — that's a*d?? NO: a*d = 1 * (1+γ²) — my calc starts (1:ℝ)*|x|² = |x|² — WRONG target. The goal after div_le_div_iff₀ is `1 * (1 + ρ.1.im ^ 2) ≤ (2 * (1 + t) ^ 2) * |ρ.1.im - t| ^ 2` — and h4 says 1+γ² ≤ 2(1+t)²·x² — directly: `calc (1:ℝ) * (1+γ²) = 1+γ² := by ring; _ ≤ 2(1+t)²x² := h4; _ = (2(1+t)²)*x² := by ring`. Let me fix the h10 block: \n\n```\n have h10 : 1 / |ρ.1.im - t| ^ 2 ≤ (2 * (1 + t) ^ 2) / (1 + ρ.1.im ^ 2) := by\n rw [div_le_div_iff₀ (by positivity : (0 : ℝ) < 1 + ρ.1.im ^ 2)\n (by positivity : (0 : ℝ) < |ρ.1.im - t| ^ 2)]\n calc (1 : ℝ) * (1 + ρ.1.im ^ 2) = 1 + ρ.1.im ^ 2 := by ring\n _ ≤ 2 * (1 + t) ^ 2 * |ρ.1.im - t| ^ 2 := h4\n _ = (2 * (1 + t) ^ 2) * |ρ.1.im - t| ^ 2 := by ring\n```\n— wait but which is c*b: c = 2(1+t)², b = |x|²: c*b = (2(1+t)²)·|x|² ✓ and a*d = 1·(1+γ²) ✓. The calc's last step: 2(1+t)²·|x|² = (2(1+t)²)·|x|² ✓ ring-equal (same thing) — actually the middle and last are syntactically identical (2*(1+t)^2*|x|^2 vs (2*(1+t)^2)*|x|^2) — ring can close or just drop the last step. Let me fix via Edit.","signature":""},{"type":"text","text":"Fixing the broken h10 calc block before compiling."},{"type":"tool_use","id":"Edit_150","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean","old_string":" have h10 : 1 / |ρ.1.im - t| ^ 2 ≤ (2 * (1 + t) ^ 2) / (1 + ρ.1.im ^ 2) := by\n rw [div_le_div_iff₀ (by positivity : (0 : ℝ) < 1 + ρ.1.im ^ 2)\n (by positivity : (0 : ℝ) < |ρ.1.im - t| ^ 2)]\n calc (1 : ℝ) * |ρ.1.im - t| ^ 2 = |ρ.1.im - t| ^ 2 := by ring\n _ ≤ 1 + ρ.1.im ^ 2 := by\n nlinarith [sq_nonneg ρ.1.im]\n _ ≤ 2 * (1 + t) ^ 2 * |ρ.1.im - t| ^ 2 := h4\n _ = (2 * (1 + t) ^ 2) * (1 + ρ.1.im ^ 2) - (2 * (1 + t) ^ 2) * ρ.1.im ^ 2\n + ((2 * (1 + t) ^ 2) * ρ.1.im ^ 2 - (2 * (1 + t) ^ 2) * |ρ.1.im - t| ^ 2)\n + (2 * (1 + t) ^ 2) * |ρ.1.im - t| ^ 2 := by ring","new_string":" have h10 : 1 / |ρ.1.im - t| ^ 2 ≤ (2 * (1 + t) ^ 2) / (1 + ρ.1.im ^ 2) := by\n rw [div_le_div_iff₀ (by positivity : (0 : ℝ) < 1 + ρ.1.im ^ 2)\n (by positivity : (0 : ℝ) < |ρ.1.im - t| ^ 2)]\n calc (1 : ℝ) * (1 + ρ.1.im ^ 2) = 1 + ρ.1.im ^ 2 := by ring\n _ ≤ 2 * (1 + t) ^ 2 * |ρ.1.im - t| ^ 2 := h4"},"extra_content":null}]