[{"type":"thinking","thinking":"Three remaining:\n\n1. **Line 1296 (hlog49's ring)**: goal `2 * Real.log 2 - Real.log 9 = Real.log 2 - Real.log 9 + Real.log 2` — wait, this IS a ring identity: 2L − M = L − M + L ⟺ 2L = 2L ✓ — `ring` FAILED?! — \"Try this: ring_nf\" — ring couldn't close `2L − M = L − M + L`?? — that's trivially true by commutativity of addition (2L − M = L + L − M = L − M + L ✓). ring should handle... — UNLESS the issue is the sides aren't parsed as ring expressions with Real.log 2/9 as atoms — ring treats them as atoms ✓ should work. Hmm — \"unsolved goals ⊢ 2 * Real.log 2 - Real.log 9 = Real.log 2 - Real.log 9 + Real.log 2\" — hmm — maybe `ring` was never called (my edit's `push_cast; ring` — the `show Real.log 4 = 2·log2 by rw [...]; push_cast; ring` — the push_cast might have closed it, ring errored?? — no — the goal shown IS this equation — the `ring` at 1296:75 FAILED to close it. Why?? — `2 * Real.log 2 - Real.log 9 = Real.log 2 - Real.log 9 + Real.log 2` — as atoms {A = log 2, B = log 9}: 2A − B = A − B + A ⟺ 2A − B = 2A − B ✓ — ring MUST close this. Unless... the `ring` at 1296:75 is in the `show Real.log 4 = 2 * Real.log 2 by ...` block — the goal after `rw [show (4:ℝ) = 2^2 by norm_num, Real.log_pow]; push_cast` — Real.log_pow gives `Real.log (2^2) = ↑2 · Real.log 2` (n : ℕ cast!) — the ↑2 is a NAT CAST — so the goal: `↑2 · Real.log 2 = 2 · Real.log 2`?? no wait — the show's statement is `Real.log 4 = 2 * Real.log 2` — after rw [show 4 = 2², Real.log_pow]: `↑2 · Real.log 2 = 2 · Real.log 2` — push_cast normalizes ↑2 → 2: `2 · Real.log 2 = 2 · Real.log 2` ✓ push_cast closes?? then ring has no goals → \"No goals\"... but the error shows the goal `2·log2 − log9 = log2 − log9 + log2` — that's the MAIN hlog49 goal (not the inner show's) — the inner show's rw chain: `rw [Real.log_div ..., show Real.log 4 = 2·log2 by ...]` — the inner show FAILED? and the main goal after rw [Real.log_div] is `log 4 − log 9 = log 2 − log 9 + log 2` with the inner show `log 4 = 2 log 2` rewriting log 4 → 2 log 2: `2·log2 − log9 = log2 − log9 + log2` ✓ — the `ring` at 1296:75 — hmm — 1296:75 col 75 — that's the MAIN hlog49's `ring`?? — wait, looking at my hlog49 edit:\n\n```\n  have hlog49 : Real.log (4 / 9) = Real.log 2 - Real.log 9 + Real.log 2 := by\n    rw [Real.log_div (by norm_num : (4 : ℝ) ≠ 0) (by norm_num : (9 : ℝ) ≠ 0),\n      show Real.log 4 = 2 * Real.log 2 by\n        rw [show ((4 : ℝ)) = 2 ^ 2 by norm_num, Real.log_pow]\n        push_cast\n        ring]\n```\nThe `push_cast; ring` inside the inner show — if push_cast closes the inner show, `ring` errors (\"No goals\") — but the error at 1296 shows the MAIN goal unsolved with the inner show unapplied?? — no wait — the displayed goal IS the main goal's final form `2·log2 − log9 = log2 − log9 + log2` — and hlog49's rw chain ENDS with the inner show rewrite — after `rw [Real.log_div ..., show ...]`, the main goal becomes `2·log2 − log9 = log2 − log9 + log2` — IDENTICAL to its RHS?? — the main goal IS `2·log2 − log9 = log2 − log9 + log2` — and `ring` should close — but 1296:75 \"unsolved goals\" with that exact goal — the `ring` FAILED on a trivially-true equation?!?! \n\nOH WAIT — I see — my edit REMOVED the final `ring` from hlog49 (the edit's rw chain ends with the inner show's `ring`, no main `ring`!). Looking at my edit: the rw chain is `rw [Real.log_div ..., show Real.log 4 = 2·log2 by rw [...]; push_cast; ring]` — after this rw, the main goal is `2·log2 − log9 = log2 − log9 + log2` — rw closes rfl-goals automatically ONLY if the resulting goal is rfl — `2·log2 − log9 = log2 − log9 + log2` is NOT rfl (different normal forms: 2A−B vs A−B+A — these ARE different syntactically!). So the main goal needs a final `ring` — which my edit DROPPED. Add `ring` after the rw chain:\n```\n  have hlog49 : Real.log (4 / 9) = Real.log 2 - Real.log 9 + Real.log 2 := by\n    rw [Real.log_div (by norm_num : (4 : ℝ) ≠ 0) (by norm_num : (9 : ℝ) ≠ 0),\n      show Real.log 4 = 2 * Real.log 2 by\n        rw [show ((4 : ℝ)) = 2 ^ 2 by norm_num, Real.log_pow]\n        push_cast\n        ring]\n    ring\n```\nHmm — but if the inner push_cast closes and its `ring` errors... — the inner show: after `rw [show ((4:ℝ)) = 2^2 by norm_num, Real.log_pow]`, goal: `↑(2:ℕ)·Real.log 2 = 2·Real.log 2` — push_cast normalizes ↑2: closes ✓ (or leaves `2·log2 = 2·log2` rfl-closed by push_cast's normalization) — then `ring` — if push_cast closed, ring errors. The error at 1296 shows the MAIN goal unsolved — so the inner show PASSED (its push_cast+ring worked) and the main rw chain left the final equation needing ring ✓ just add it.\n\n2. **Line 1277 (the identity's final ring)**: goal `Real.log 3 * 2 - Real.log 3 * √↑n * 2 = -(Real.log 3 * √↑n * 2) + Real.log 9` — with h9 : log 9 = 2 log 3 — but h9 was supposed to be APPLIED (rw [h9]) — the goal still has `Real.log 9` — so rw [h9] didn't fire (h9's rewrite happened BEFORE the mul_assoc/← Real.exp_add steps... — looking at my edited chain: `rw [hlog49, h9, mul_assoc, ← Real.exp_add]` — rw [h9] should rewrite log 9 → 2 log 3 EVERYWHERE — the displayed goal HAS `Real.log 9` — so h9 didn't rewrite?! — OH WAIT — looking at my edit again: the chain is `rw [hlog49, h9, mul_assoc, ← Real.exp_add]` — hmm, `mul_assoc` — rw [mul_assoc] rewrites (a·b)·c → a·(b·c) EVERYWHERE (the session-033 gotcha: \"rw [X = 1*X] rewrites everywhere\") — mul_assoc might have rewritten the WRONG product or created a mess. And the final ring's goal has log 9 STILL there — h9 was applied at an earlier stage and the final exponent has `...+ Real.log 9` — the goal: `Real.log 3 * 2 - Real.log 3 * √↑n * 2 = -(Real.log 3 * √↑n * 2) + Real.log 9` — with log 9 = 2 log 3: RHS = −2·log3·√↑n + 2·log3 = 2·log3·(1−√↑n)·... = log3·(2 − 2√↑n) = −log3·(2√↑n−2)·... and LHS = log3·(2 − 2√↑n) ✓ SAME ✓ — the ring needs log 9 → 2 log 3 — h9 was already applied EARLIER but the final `show (9:ℝ) = exp (log 9)` rewrite REINTRODUCED log 9 (via the exp_log)! Then the final ring needs log 9 → 2 log 3 again — rw [h9] before the final ring. Looking at my chain: `rw [show (9:ℝ) = Real.exp (Real.log 9) from (Real.exp_log ...).symm]; rw [← Real.exp_add]; congr 1; ring` — after the show, `exp(log 9)` appears; ← Real.exp_add combines `exp(log 9)·exp(A+B)` → `exp(log 9 + (A+B))`; congr 1: exponents equal: `−log3·(2√↑n−2) = log 9 + (A+B)` — the ring needs h9 again: the displayed goal `Real.log 3 * 2 - Real.log 3 * √↑n * 2 = -(Real.log 3 * √↑n * 2) + Real.log 9` — ring with h9: `2·log3·(1−√↑n) = −2·log3·√↑n + 2·log3` ✓ — but ring doesn't know h9. Fix: `rw [h9]; ring` at the end — but h9 : log 9 = 2 log 3 — rw [h9] rewrites log 9 → 2 log 3 ✓ then ring ✓.\n\nWait — but the goal display at 1277 shows `Real.log 3 * 2 - Real.log 3 * √↑n * 2 = -(Real.log 3 * √↑n * 2) + Real.log 9` — with log 9 present — so the chain's earlier rw [h9] was applied but the show's exp_log reintroduced a FRESH log 9 that h9 (applied before the show) didn't touch. So add `rw [h9]` before the final ring:\n```\n  rw [show (9 : ℝ) = Real.exp (Real.log 9) from (Real.exp_log (by norm_num : (0 : ℝ) < 9)).symm]\n  rw [← Real.exp_add, h9]\n  congr 1\n  ring\n```\nHmm — order: after the show, the goal has exp(log 9) — rw [← Real.exp_add] combines exp(log 9)·exp(A+B)?? — wait, the goal's RHS after my earlier steps is `9 * Real.exp ((...)) * Real.exp (−log2·(2√↑n))` — after mul_assoc: `9 * (Real.exp A * Real.exp B)` — ← Real.exp_add: `9 * Real.exp (A+B)` — then the show: `Real.exp (log 9) * Real.exp (A+B)` — ← Real.exp_add: `Real.exp (log 9 + (A+B))` — then rw [h9] (log 9 → 2 log 3) then congr 1; ring. Let me restructure the identity's tail completely (the current version is tangled). The goal: `exp(−log3·(2√↑n−2)) = 9·exp(A)·exp(B)` with A = (log(4/9))·√↑n... — hmm wait — let me re-derive what the identity proof should be:\n\nGoal: 3^{−(2√↑n−2)} = 9·(4/9)^{√↑n}·2^{−2√↑n}.\nLHS in exp form: exp(−log3·(2√↑n−2)) = exp(−2·log3·√↑n + 2·log3) = exp(2·log3)·exp(−2·log3·√↑n) = 9·exp(−2·log3·√↑n) (exp(2·log3) = (exp log 3)² = 9).\nRHS: 9·exp(log(4/9)·√↑n)·exp(−log2·(2√↑n)) — log(4/9)·√↑n − 2·log2·√↑n = √↑n·(log(4/9) − 2 log 2) = √↑n·(2log2 − log9 − 2log2) = −√↑n·log9 = −2·log3·√↑n ✓.\nSo LHS = 9·exp(−2·log3·√↑n) = RHS = 9·exp(log(4/9)·√↑n − 2·log2·√↑n) ✓.\n\nThe cleanest Lean proof:\n```\nlemma three_pow_neg_two_sqrt_sub_two_eq {n : ℕ} :\n    (3 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) =\n      9 * ((4 / 9 : ℝ) ^ Real.sqrt (n : ℝ)) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ))) := by\n  have h2 : (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ))) =\n      Real.exp (-(Real.log 2) * (2 * Real.sqrt (n : ℝ))) := by\n    rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2)]\n    congr 1\n    ring\n  have h3 : (3 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) =\n      Real.exp (-(Real.log 3) * (2 * Real.sqrt (n : ℝ) - 2)) := by\n    rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 3)]\n    congr 1\n    ring\n  have h49 : (4 / 9 : ℝ) ^ Real.sqrt (n : ℝ) =\n      Real.exp (Real.log (4 / 9) * Real.sqrt (n : ℝ)) :=\n    Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 4 / 9) _\n  have h9 : Real.log 9 = 2 * Real.log 3 := by\n    rw [show (9 : ℝ) = 3 ^ 2 by norm_num, Real.log_pow]\n    push_cast\n    ring\n  have hlog49 : Real.log (4 / 9) = -2 * Real.log 3 + Real.log 2 := by\n    rw [Real.log_div (by norm_num : (4 : ℝ) ≠ 0) (by norm_num : (9 : ℝ) ≠ 0),\n      show Real.log 4 = 2 * Real.log 2 by\n        rw [show ((4 : ℝ)) = 2 ^ 2 by norm_num, Real.log_pow]\n        push_cast\n        ring, h9]\n    ring\n  rw [h2, h3, h49, hlog49]\n  rw [← Real.exp_add, ← Real.exp_add]\n  have h9exp : (9 : ℝ) = Real.exp (2 * Real.log 3) := by\n    rw [← h9, Real.exp_log (by norm_num : (0 : ℝ) < 9)]\n  rw [h9exp, ← Real.exp_add]\n  congr 1\n  ring\n```\nHmm — hlog49 : log(4/9) = −2·log3 + log2 (avoiding the log 9 in the statement — cleaner for the ring finish). Then after rw [h2, h3, h49, hlog49]: the goal is `exp(−log3·(2√↑n−2)) = 9·exp((−2log3+log2)·√↑n)·exp(−log2·(2√↑n))` — ← Real.exp_add twice: first combines exp(A)·exp(B) → exp(A+B)?? — the RHS `9 * exp(A) * exp(B)` — with mul_assoc needed... — hmm, `9 * exp(A) * exp(B)` = `(9·exp(A))·exp(B)` — ← Real.exp_add's pattern `exp ?x * exp ?y` — the subterm `exp(A)·exp(B)` — is `(9·exp(A))·exp(B)` containing it? — NO (the 9 is glued to exp A). So: `rw [mul_assoc]` first to get `9·(exp A · exp B)`, then `← Real.exp_add`: `9·exp(A+B)`, then h9exp : 9 = exp(2·log3): rw → `exp(2·log3)·exp(A+B)`, then `← Real.exp_add`: `exp(2·log3+(A+B))`, then congr 1: exponent identity `−log3·(2√↑n−2) = 2·log3 + (A+B)` where A = (−2·log3+log2)·√↑n, B = −log2·(2√↑n): A+B = √↑n·(−2·log3+log2−2·log2)·... = √↑n·(−2·log3−log2)... hmm let me verify: A = (−2log3+log2)·√↑n; B = −2·log2·√↑n; A+B = (−2log3+log2−2log2)·√↑n = (−2log3−log2)·√↑n. And LHS exponent: −log3·(2√↑n−2) = −2·log3·√↑n+2·log3. And the combined RHS exponent after h9exp: 2·log3 + (A+B) = 2·log3 + (−2·log3−log2)·√↑n — hmm: 2·log3 + (−2·log3−log2)·√↑n vs −2·log3·√↑n + 2·log3 — DIFFERENT by the −log2·√↑n term!! Let me recompute B: B = −(log 2)·(2·√↑n) = −2·log2·√↑n ✓. A+B = (−2·log3+log2)·√↑n + (−2·log2·√↑n) = (−2·log3+log2−2·log2)·√↑n = (−2·log3−log2)·√↑n?? — that has a −log2·√↑n term that shouldn't be there...\n\nLet me recompute the TARGET identity: 3^{−(2√↑n−2)} = 9·(4/9)^{√↑n}·2^{−2√↑n}.\nlog 3^{−(2√↑n−2)} = −(2√↑n−2)·log 3 = −2√↑n·log3 + 2·log3.\nlog RHS = log 9 + √↑n·log(4/9) − 2√↑n·log 2 = 2·log3 + √↑n·(2·log2 − 2·log3) − 2√↑n·log2 = 2·log3 + 2√↑n·log2 − 2√↑n·log3 − 2√↑n·log2 = 2·log3 − 2√↑n·log3 ✓✓ EQUAL ✓.\n\nSo the exponents: LHS = −2·log3·√↑n + 2·log3; RHS = 2·log3 + √↑n·(2·log2−2·log3) + (−2·log2·√↑n) = 2·log3 + √↑n·(2·log2−2·log3−2·log2) = 2·log3 − 2√↑n·log3 ✓ ✓.\n\nSo in the Lean proof: A = log(4/9)·√↑n (with log(4/9) = 2·log2 − log9 = 2·log2 − 2·log3); B = −log2·(2·√↑n) = −2·log2·√↑n. A+B = (2·log2−2·log3)·√↑n + (−2·log2·√↑n) = −2·log3·√↑n ✓✓ — I had an arithmetic slip before. Good — so hlog49 : log(4/9) = 2·log2 − 2·log3 ✓ (with h9 : log 9 = 2 log 3 used). And the final ring: 2·log3 + (A+B) = 2·log3 + (−2·log3·√↑n) — and LHS exponent −2·log3·√↑n+2·log3 ✓ EQUAL ✓ ring closes ✓ (it's the SAME normal form? — LHS: −(log3·(2√↑n−2)) = −2·log3·√↑n+2·log3; RHS: 2·log3+(−2·log3·√↑n) = 2·log3−2·log3·√↑n ✓ — ring normalizes both to 2·log3−2·log3·√↑n ✓ closes ✓).\n\nLet me rewrite the whole identity lemma cleanly. And error 3 (1332) is a cascade — will resolve once these are fixed.","signature":""},{"type":"text","text":"The identity proof is tangled — rewriting it cleanly (with the exponent arithmetic worked out)."},{"type":"tool_use","id":"Read_480","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","offset":1276,"limit":42},"extra_content":null}]