← Session 29 — ii′-5e-v: circle bound (far-tail + series bounds)
thinking (1549 chars)
Line 3275: `linarith [h19]` — goal: `s.re / 2 + R / 2 < 50 / R` from h19 : `R + s.re < 100 / R`. s.re/2+R/2 = (R+s.re)/2 < (100/R)/2 = 50/R ✓ linear in the atom (100/R) vs (50/R): linarith needs (100/R)/2 = 50/R — DIVISION atoms again! `100 / R` and `50 / R` are different opaque atoms and linarith doesn't know 100/R = 2·(50/R). Banked gotcha (the P=2Q pattern)! Fix: rewrite the goal: `show s.re / 2 + R / 2 < 100 / R / 2` then linarith [h19] — is 100/R/2 the same atom as in h19? h19 has `100 / R`: goal `100 / R / 2` = (100/R)/2 ✓ same atom. So:
```
have hlt : s.re / 2 + R / 2 < 50 / R := by
...
have h19 : R + s.re < 100 / R := ...
have h20 : s.re / 2 + R / 2 = (R + s.re) / 2 := by ring
rw [h20]
-- goal: (R + s.re)/2 < 50/R: from h19: (R+s.re)/2 < (100/R)/2: bridge 50/R = (100/R)/2:
have h21 : (50 : ℝ) / R = 100 / R / 2 := by field_simp; ring
rw [h21]
linarith [h19]
```
Or simpler: state h19 as `R + s.re < 2 * (50 / R)` hmm — cleaner: h19's form: R + s.re < 100/R: then goal (R+s.re)/2 < 50/R ⟺ R+s.re < 2·(50/R): linarith [h19] needs 100/R = 2·(50/R) — same atom problem. Do the h21 bridge.
Similarly — the h50 : 50/R ≤ 1/8 ✓ fine (constants).
Also the final linarith [h1, h4, hgap', h9, hlt, h50]: chain: |s.re/2+m| ≥ |m−R/2| − (s.re/2+R/2): h4; hgap' : 1/4 ≤ |R/2−m| with h9 : |R/2−m| = |m−R/2|: so |m−R/2| ≥ 1/4; hlt : s.re/2+R/2 < 50/R; h50 : 50/R ≤ 1/8: so |s.re/2+m| > 1/4 − 1/8 = 1/8: goal 1/8 ≤ ‖s/2+m‖ with h1 : ‖s/2+m‖ ≥ |s.re/2+m| ✓ linear ✓.
Let me fix both spots (div_lt_iff₀ form + the hlt atom bridge).
Two atom-bridge fixes: `div_lt_iff₀`'s left-mult form and the 100/R = 2·(50/R) bridge.
tool: Edit
input
{
"replace_all": false,
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean",
"old_string": " have h6 : (R - |s.re|) * R < 100 := by linarith [h3, h5]\n exact (div_lt_iff\u2080 hRpos).mpr h6",
"new_string": " have h6 : (R - |s.re|) * R < 100 := by linarith [h3, h5]\n have h7 : R * (R - |s.re|) = (R - |s.re|) * R := by ring\n exact (div_lt_iff\u2080 hRpos).mpr (by rw [h7]; exact h6)"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Line 3275: `linarith [h19]` — goal: `s.re / 2 + R / 2 < 50 / R` from h19 : `R + s.re < 100 / R`. s.re/2+R/2 = (R+s.re)/2 < (100/R)/2 = 50/R ✓ linear in the atom (100/R) vs (50/R): linarith needs (100/R)/2 = 50/R — DIVISION atoms again! `100 / R` and `50 / R` are different opaque atoms and linarith doesn't know 100/R = 2·(50/R). Banked gotcha (the P=2Q pattern)! Fix: rewrite the goal: `show s.re / 2 + R / 2 < 100 / R / 2` then linarith [h19] — is 100/R/2 the same atom as in h19? h19 has `100 / R`: goal `100 / R / 2` = (100/R)/2 ✓ same atom. So:\n```\nhave hlt : s.re / 2 + R / 2 < 50 / R := by\n ...\n have h19 : R + s.re < 100 / R := ...\n have h20 : s.re / 2 + R / 2 = (R + s.re) / 2 := by ring\n rw [h20]\n -- goal: (R + s.re)/2 < 50/R: from h19: (R+s.re)/2 < (100/R)/2: bridge 50/R = (100/R)/2:\n have h21 : (50 : ℝ) / R = 100 / R / 2 := by field_simp; ring\n rw [h21]\n linarith [h19]\n```\nOr simpler: state h19 as `R + s.re < 2 * (50 / R)` hmm — cleaner: h19's form: R + s.re < 100/R: then goal (R+s.re)/2 < 50/R ⟺ R+s.re < 2·(50/R): linarith [h19] needs 100/R = 2·(50/R) — same atom problem. Do the h21 bridge.\n\nSimilarly — the h50 : 50/R ≤ 1/8 ✓ fine (constants).\n\nAlso the final linarith [h1, h4, hgap', h9, hlt, h50]: chain: |s.re/2+m| ≥ |m−R/2| − (s.re/2+R/2): h4; hgap' : 1/4 ≤ |R/2−m| with h9 : |R/2−m| = |m−R/2|: so |m−R/2| ≥ 1/4; hlt : s.re/2+R/2 < 50/R; h50 : 50/R ≤ 1/8: so |s.re/2+m| > 1/4 − 1/8 = 1/8: goal 1/8 ≤ ‖s/2+m‖ with h1 : ‖s/2+m‖ ≥ |s.re/2+m| ✓ linear ✓.\n\nLet me fix both spots (div_lt_iff₀ form + the hlt atom bridge).","signature":""},{"type":"text","text":"Two atom-bridge fixes: `div_lt_iff₀`'s left-mult form and the 100/R = 2·(50/R) bridge."},{"type":"tool_use","id":"Edit_189","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":" have h6 : (R - |s.re|) * R < 100 := by linarith [h3, h5]\n exact (div_lt_iff₀ hRpos).mpr h6","new_string":" have h6 : (R - |s.re|) * R < 100 := by linarith [h3, h5]\n have h7 : R * (R - |s.re|) = (R - |s.re|) * R := by ring\n exact (div_lt_iff₀ hRpos).mpr (by rw [h7]; exact h6)"},"extra_content":null}]